Let $R$ be a commutative unital ring and $S$ be a normal subgroup of a finite group $G$. If $P$ is a finitely generated projective left $R[G]$-module then is the submodule of $S$-invariants $$P^S:= \{p\in P:g(p)=p\ \forall g\in S\}$$ a projective left $R[G/S]$-module?
We know that there is a $R[G]$-module $Q$ and a positive integer $n$ s.t. $P\oplus Q\cong R[G]^n$. So $P^S\oplus Q^S=(P\oplus Q)^S\cong (R[G]^n)^S\cong (R[G]^S)^n.$
Now is it true that $R[G]^S\cong R[G/S]?$ I'm not sure how to show this or if this is the way to proceed.
Also is this result true if $P$ is not finitely generated?
Many thanks for your help.