Question

Can someone provide a reference for the following Maschke lemma:

If $C$ is a semisimple algebra over a field of characteristic zero and $G$ is a finite group acting on $C$ by automorphisms, then $C \rtimes G$ is semisimple.

Answer 1

Theorem 7.5.6(iii) of the book Noncommutative Noetherian Rings by J.C.McConnell and J.C.Robson states the following:

Let $R$ be a ring, $G$ a finite group with $|G|$ a unit in $R$, and $S = R \ast G$, a crossed product of $R$ by $G$. Then the right global dimension of $R$ equals the right global dimension of $S$.

An algebra is semisimple if and only if it has right (or left) global dimension zero, and your skew group ring $C \rtimes G$ is a special case of the more general crossed product construction. So your Maschke lemma is a special case of the result cited above.