Let $X$ be scheme and $G \subset Aut(X)$ be a subgroup of automorphism group of $X$. By definition $G$ acts espectially on local sections $\mathcal{O}_X(U)$ for open $U$ and one can therefore define the subsheaf $\mathcal{O}_X^G$ of $G$-invariants on $X$.
My question is if and how there exist a canonical way to extend the action of $G$ to all coherent sheaves $\mathcal{F}$ over $X$ or resp $\mathcal{O}_X$ modules?
Background: How to define the sub sheaves of invariants $\mathcal{F}^G$ for arbitrary $\mathcal{F}$?
If yes, does $G$ induce a functor $^G$ on the category of sheaves over $X$? When is it exact?
Ok, this question makes more sense to me than your other (earlier) question.
$G$ doesn't act on local sections. $G$ isn't guaranteed to preserve open sets, (i.e. $gU\ne U$ necessarily) so how could it possibly act on $\newcommand\calO{\mathcal{O}}\calO_X(U)$ for all open $U$?
Instead, if $U$ is $G$-invariant, $G$ descends to an action on $\calO_X(U)$ in the obvious way. Note, however that $G$ acts by ring homomorphisms not module morphisms.
In general, the action of $G$ on $X$ will not induce an action of $G$ on a particular $\calO_X$-module. For example, if we take a sky scraper sheaf over some point with nontrivial orbit, then clearly $G$ will act by sending the sky scraper sheaf to a sky scraper sheaf with a different base point.