Group action on ringed space

Question

I am trying to understand the action of a finite group on a ringed space. let $G$ be a finite group acting on $(X, O_X)$. I know that $g \in G $ induces a morphism $g: X\longrightarrow X$ and for the open set $U\subseteq X$, we have $g^\#: \mathcal{O}_X(U)\longrightarrow \mathcal{O}_X(g^{-1}(U))$. If we have a quotient $f: X\longrightarrow Y = X/G$, is there anybody who help me to understand how $G$ acts on the $\mathcal{O}_X(f^{-1}(V))$ for open $V\subseteq Y$? How can I show $f^{-1}(V)$ is G-invaraint(is it realy)?