Let $f:R\rightarrow S$ be a homomorphism of (commutative) rings with unity and suppose that $G$ is a group acting on $R$ and $S$ in such a way that $f\sigma=\sigma f$ for every $\sigma\in G$. Denote by $R^{G}$ and $S^{G}$ the respective rings of invariants of $G$, this is, $R^{G}=\left\{r\in R:\sigma r=r\textrm{ for every }\sigma\in G\right\}$ and $S^{G}=\left\{s\in R:\sigma s=s\textrm{ for every }\sigma\in G\right\}$. Then it is induced a ring homomorphism $f^{G}:R^{G}\rightarrow S^{G}$. Suppose also that $S$ is a projective $R$-module. Is it true that $S^{G}$ is a projective $R^{G}$-module?

Thank you.