Is the speed of light in a medium also remains constant when the medium travels at high speed? There is a problem given as

A light signal travels from $A$ to $B$ within a glass slab which is moving (in same direction as that of light) with speed 0.3c with respect to an external observer.If the refractive index of slab is 1.5, then what is the speed of the signal as observered by the observer?

The problem is easy. We know that
$$\begin{align}
\text{Velocity of light in a medium} \,\,=&\,\, \frac{\text{Velocity of light in vaccuum}}{\text{Refractive index of the material}}\\\text{Velocity of light in glass slab} \,\,=&\,\, \frac{c}{1.5}\end{align}$$
This velocity of signal comes out in the frame in which the glass slab is stationary. But as the signal is light signal, so I think that the speed of signal will also appear same to the observer for whom the glass slab appears to move.
But in the solution, the relativistic addition of velocity is also used ,i.e.,
$$\displaystyle u_x=\frac{u_x'+v_{frame}}{1+\frac{u_x'v_{frame}}{c^2}}$$
$$\displaystyle\implies u_x=\frac{\frac{c}{1.5}+0.3c}{1+\frac{0.3}{1.5}}=0.81c$$
This is the final answer given.
But I have a doubt that in relativity it is assumed that speed of light in vaccuum remains constant in all inertial frames. But doesn't then speed of light in any medium should also remain constant in any other inertial frame (irrespective of material of those other frames)?
Please explain why velocity transformation equations are used after getting speed of light in the frame of glass slab. The observer should also observe the velocity of light to be $\frac{c}{1.5}$.
Please clasrify the doubt!
 A: What is a 'medium'? It involves matter. In particular atoms which interact with light. The light speed within the medium will thus depend on many properties of the medium, in the particular the interatomic distance and rate of absorption/emission. Both these properties change when changing reference frame due to time dilation and length contraction. In other words, if you know a priori the interatomic distance of your material and you now measure it to be smaller you find out that there is relative motion between you and the medium, therefore you can distinguish the two frames, so there's no reason to think that the speed of light should remain invariant. This is different from being in a vacuum, where you can't just tell the speed you're moving, and thus speed of light should be the same in any frame.
A: The speed of light in a medium does depend on the relative velocity of the medium and observer. This in fact was experimentally observed even before the theory of relativity was created, and was one of the results that influenced Einstein's thinking. The experiment was first performed by Fizeau in 1851, and has been repeated many times since. Fizeau's results were surprising to 19th century physicists because they could not be explained by any simple velocity addition formula; they are, however, explained by the relativistic velocity addition formula.
