This is a basic question (not research level) which has already been asked on SE by someone else but doesn't yet have an answer so I'd like to repost it on MO.

Let $R$ be a commutative ring with unity and $H$ 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 $H$-invariants $$P^H:= \{p\in P:g(p)=p\ \forall g\in H\}$$ a projective left **$R[G/H]$**-module?

There is a $R[G]$-module $Q$ and a positive integer $n$ s.t. $P\oplus Q\cong R[G]^n$. So $P^H\oplus Q^H=(P\oplus Q)^H\cong (R[G]^n)^H\cong (R[G]^H)^n.$

Is it true that $R[G]^H\cong R[G/H]?$ Or is there is another way to proceed?

Also is this result true if $P$ is not finitely generated?

Many thanks.