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.
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.
This is also stated in the generality asked for as Theorem 1.3 (c) in [Reiten, Riedtmann: Skew group algebras in the representation theory of Artin algebras, Journal of Algebra 92, 1985].