I am currently reading D. Mumford´s Abelian Varieties and it came up the following question: let $X$ be an algebraic variety over an algebraically closed field $k$ and $G$ a finite group acting on $X$. Assume we are in a situation that there exists a quotient $(Y, \pi:X \rightarrow Y)$, he then proves a proposition that there is a one-to-one correspondence between coherent $\mathcal{O}_{Y}$-modules and $G$-equivariant coherent $\mathcal{O}_{X}$-modules. In the course of the proof he shows that for such a $G$-sheaf $\mathfrak{F}$ on $X$ the natural morphism $\pi^{-1}((\pi_{*}\mathfrak{F})^{G}) \rightarrow \mathfrak{F}$ is an isomorphism.

What about other kind of sheaves, especially locally constant sheaves (regarding whether this natural morphism is an isomorphism)?