Does induced current affect the permeability of a material? For electromagnetic wave propagation, we can typically characterize a medium by conductivity $\sigma$, permeability $\mu$, and permittivity $\epsilon$. Consider if this material had current passing through it. Would it possibly change the permeability $\mu$? So if you had a rod of mu-metal for example, you know the permeability $\mu$ but does that change if you induce current through the rod?
Update: So I understand that B-H the relationship has a point where the material saturates, and increasing the incident field only causes a small increase in the induced field, and effectively reduces permeability $\mu$. However, in the above example, assume that the incident H field is small enough that it does not reach saturation. What I am talking about is if you connected wires at the 2 ends of the rod and connected it to a current source (AC or DC), does that change the permeability $\mu$?
Perhaps you guys already understood this but let me know if your answer changes.
 A: Absolutely. Saturation of the magnetic material by DC currents/magnetic fields is a primary concern in power electronics, where it can be one of the primary failure mechanisms, if the designer underestimates the significant drop in permeability, which leads to a drop in inductance. Usually the DC magnetic field is not caused by current trough the magnetic material itself, but by current trough a coil around it, but the effects would be the same. 
The effect is also being used for extremely sensitive magnetic field sensors. Take a look at fluxgate magnetometers, which despite their very simple and rugged design can achieve amazing performance: http://www3.imperial.ac.uk/spat/research/areas/space_magnetometer_laboratory/spaceinstrumentationresearch/magnetometers/fluxgatemagnetometers
Another application would be magnetic amplifiers, although they are rarely used today. 
The general rule for using magnetic materials is, that one has to carefully understand the magnetic environment and the current flowing trough them, otherwise very unexpected, and in many cases undesired events may happen, especially with high permeability materials like mu-metal. 
A: The answer therefore is a qualified No.  A current passing through a material will create a field, which will certainly affect the magnetic permeability $\mu$ if the material is magnetically nonlinear.  If the material is mu-metal then it will saturate at a low field.

So if you had a rod of mu-metal for example, you know the permeability μ but does that change if you induce current through the rod?

If you induce a current then you must be applying a changing magnetic field, which will certainly affect the nonlinear material, even if the induced current itself does not create a significant field.
However, I don't know of any mechanisms by which an electric current itself can affect the permeability - but there is a big zoo of effects out there!  Since magneto-resistance exists, I suppose that some sort of inverse effect might well exist whereby a current could affect permeability. 
