Can inflation extract information from black holes? Suppose inflation were to occur for some brief period of time within the Schwarzschild radius of a black hole such that some matter which is just inside the event horizon now finds itself outside of the Schwarzschild radius.
Wouldn't this mean we can get information from the matter and energy that was previously behind the event horizon?
Did such a thing not occur during the inflationary epoch to literally all matter that is now not inside primordial black holes?
 A: PNS already pointed out that retrieving information/matter from beyond the black hole's (BH) event horizon is not possible. Let me answer from a different perspective (summarizing my comments).
Event horizon of a black hole is DEFINED as a region surrounding its singularity* from which you cannot retrieve information. If the expansion of space manages to expel an object/particle from the event horizon, this would mean there was no event horizon to begin with (by definition!). In this situation you would have naked singularity of some sort, and now you can transform your question into "Are there naked singularity solutions in expanding space?"
Unfortunately I cannot answer this question, but I can point to the presumably right direction. Expansion of space can be due to positive cosmological constant - corresponding to de Sitter spacetime - or due to a scalar field(s) similarly to inflation. In the former case you have exponential expansion of space, meaning the scale factor is $a=e^{Ht}$ where $H=H(t)$ is Hubble function and $t$ is time. In the latter case the expansion is only approximately exponential, $a\simeq e^{Ht}$ - this is called quasi-de Sitter spacetime. On the other hand, (static) black hole in flat (Minkowski) spacetime is described by Schwarzschild solution. If you want to consider a black hole in expanding space, you have to modify the pure Schwarzschild solution to take into account the expansion of space. This would be given by the so-called Schwarzschild-de Sitter solution in the simplest case where the expansion is due to a cosmological constant.
In summary, the answer to the actual question is no, matter cannot be expelled from the event horizon by inflation. But your question can also be transformed into "Can there be naked singularities in Schwarzschild-de Sitter or similar spacetimes?"
*Instead of singularity, black hole solutions can also have smooth/nonsingular cores for certain specific cases like Bardeen black hole, see e.g. https://arxiv.org/abs/gr-qc/9911046, but for simplicity let's call it singularity. Also keep in mind that singularity is an artifact of our description of gravity (GR) that indicates breakdown of GR at microscopic scales.
Addendum.
In order for the expansion to be able to move any two objects apart from each other it needs to overcome the gravitational binding force between the objects. That would be extremely hard to do in the vicinity of a black hole let alone inside the event horizon.
A: In short: this is not going to work. Long answer: we can try and make a lot of questionable assumptions and see what comes up.

Assumption 1 - Inflation can happen inside the Schwarzschild radius:
So, we still don't know what exactly happens within a black hole at all. As someone in the comments pointed out, inflation is caused by the inflaton field, and we don't know whether it will ever be re-activated at all, let alone inside a black hole, which we know almost nothing about. So, seem highly unlikely. But, still let's assume that the inflaton field gets activated somehow, and it causes inflation inside the black hole.
Assumption 2 - Matter increases volume due to inflation:
Okay. Inflation happens to space. Space inflates when the inflaton field is activated. But, matter at the singularity is reduce to zero volume, that is infinite density. So without space to expand, how will matter go expand in volume? You may say that the Big bang (another singularity) happened. But, inflation occurred sometime after the Big bang, when it had some volume. But, for the sake of moving this ahead, let's say it can expand, somehow and see what happens.
Assumption 3 - Black holes can exist without infinite density:
The Schwarzschild metric is a solution for the equations of General Relativity, which describes the curvature of spacetime around a spherical body. It does this by using the mass of the body and the radius of the body. So, when the radius becomes zero, that is infinite density, you get a black hole. But, without the infinite density zero volume, the black hole just does not exist. It will have a strong gravitational field, but it will not be the inescapable menace we all know and love. But this the most important assumption of all.
Plot-'hole' (pun totally intended): The black hole doesn't exist now; so we get all the information out, right?
Well, we know things can become black holes, but we don't know whether the reverse is true. As I said black holes are mysterious, and we don't know much about them. So we don't think black holes can become clumps of mass, but it's unlikely. I found an answer in Astronomy SE, in which one answerer said that the black hole has a high entropy state, while clumps of matter, like stars etc. have lower entropy states. So, it is not possible for a black hole to turn back into normal matter as far as we know. Continuing on.
Assumption 4 -  Stuff can come out of the black hole:
You said in a comment that the situation is that inflation happens inside the black hole and stuff spills out. But nothing can escape the black hole. That is the thing. I don't know exactly, but most likely, inflation will just cause the space inside the black hole to expand, not throw matter out of it. Only Hawking radiation comes out of the black hole. But, let's also assume this actually would happen.
Assumption 5 - You get meaningful information from black hole insides:
This is the fun part: how are you sure that you get meaningful information from black hole? Whenever the black hole absorbs something, the area of the event horizon increase, which in itself means an increase of the entropy of the black hole. The entropy of a system corresponds to the lack of information. But, let's suppose that as matter comes out, the event horizon surface area decreases. Okay. But, how do we know that this information is really meaningful. As far as we can tell, no 'shaped stuff' exists in a black hole. If I throw in a lightsaber, then the lightsaber will not remain a lightsaber. Nobody know what else it would become. Maybe just random atoms, or perhaps due to the infinite density, just subatomic particles, or maybe even strings. I will refer to this as black hole matter.
When this matter comes out, the surface area of the black hole decreases, but do we get meaningful information about the lightsaber from this 'black hole pseudo-matter'? Probably not. You may say that the black hole surface area decreases, which represents a lowering of entropy, which should correspond to information. But, not considering all the laws of physics we are breaking, and would break; this seems totally unlikely. Event horizons lose surface areas a lot, when they release Hawking radiation, and we have no idea if they carry any information, but according to the current view, they seem to contain no information about the black hole insides.
What do I mean by meaningful information?
Meaningful information means that you get information that is helpful to you; actually makes some sense to you. If you photograph the lightsaber, and send it to me, I get meaningful information about it's size, mass, structure etc. But, throw it into  black hole and do the above experiment, and you don't get information. You can't even actually say or control what is coming out of the black hole, let alone know the original object's properties. It is something like quantum entanglement. If your friend runs the lottery, and he gets the results somehow, he can instantaneously send information to you with entangled particles. But, that entropy would make no sense to me; it would be just garbled and random information (guess you can never cheat on the lottery :-)
So, let's look at the list of assumptions:

*

*Inflation can happen inside the Schwarzschild radius.

*Matter increases volume due to inflation.

*Black holes can exist without infinite density.

*Stuff can come out of the black hole.

*You get meaningful information from black hole insides.

Some don't seem to help us much, while some just upright break the laws of physics (forbidden!). So,
CONCLUSION: You cannot get meaningful information from a black hole, because of inflation inside the Schwarzschild radius.
Appendix: What about the primordial universe?
The universe started out from a singularity, so you may argue that inflation worked then. Actually, the Big Bang worked then. The inflation epoch happened a little while after the Big Bang happened, so it did not actually have any effect on the singularity (because the singularity was now a very small universe with some volume. Albeit, small volume, it had space).
Note: This is one answer that actually helped me a lot. @Nathaniel's answer provides a huge clarification about entropy and information. If you want more detail on the connection between the two, I recommend this one: Entropy and Information.
