I am trying to find more generalized counterparts of some well-known results from modular group representations.
My question is the following:
Suppose that $H$ is a finite $p$-group acting as automorphisms on a finite dimensional $k$-algebra $A$. Suppose moreover that the field $k$ has characteristic exactly this prime $p$. Does it follow that all simple $A\#kH$ modules are in bijection with all simple $A$-modules? Obviously, I am talking about isomorphism classes of simple modules.
In the case $A=kL$ for a $p'$-group $L$ I think this result is well known although I cannot identify a reference. At least a treatment of this case can be found in Section 8.4 of Webb's book:
In this case, there is a canonical projection $A\# kH\rightarrow kH$ which makes $kH$ a canonical $A\# kH$-module. It turns out that this is a projective module and it is the projective cover of the trivial module.
Any reference to this type of smash products is very welcomed.