The motivation/background of my question arises from following thread:
Galois morphism - group acting on the variety
The original setting is that we have a finite Galois morphism $f: X \to S$, where $X$ and $S$ are non-singular and connected projective varieties over $\mathbb{C}$.
Galois means here that if we denote with $G$ the automorphism group of $X$ over $S$ then the quotient $X/G$ exists and the natural morphism $X/G \to S $ is an isomorphism.
For futher discussion I would like to use the language of schemes to consider the topic from more general point of view by taking into account that basically a variety is a scheme $X$ over $k$ such that $X$ is integral and the structure morphism $X \to Spec(k)$ is separated and of finite type.
While the discussion @user52991 mentioned following statement:
Let $$0 \to F′ \to F \to F′′\to 0$$ be a short exact sequence of coherent sheaves on X. Let the push forward $f_*$ of the short exact sequence to $S=X/G$ be exact (since $f$ finite this holds automatically). The aim is to consider the functor which associates to any coherent sheaf $E$ on $S$, its subsheaf $E^G$ of $G$ invariants. I don't know why it is well defined.
Here occure some understanding problems to me:
First of all: How does $G$ act on local sections of $X$? Namely, for $U \subset X$ the rings $\Gamma(U, \mathcal{O}_X)$?
So if we take $g \in G= Aut(X)$ then $g: X \to X$ as morphism of schemes induce ring maps
$$g_U ^{\#}:\Gamma(U, \mathcal{O}_X) \to \Gamma(U, g_*\mathcal{O}_X)= \Gamma(g^{-1}(U), \mathcal{O}_X)$$
And here occurs to me the core question how $G$ can act on the local sections $\Gamma(U, \mathcal{O}_X)$? This action only make sense if $g^{-1}(U)=U$ for all $g \in G$ and $ U \subset X$, right? Therefore I don't know why it make sense to define the invaraint subring $\Gamma(U, \mathcal{O}_X)^G$. On the other hand since $g^{-1}(X)=X$ it make sence to define the action on global sections $\Gamma(X, \mathcal{O}_X)$.
Therefore only in this special case it make sense to set $\Gamma(X, \mathcal{O}_X)^G := \Gamma(X/G, \mathcal{O}_{X/G})$. But what about local sections?
Since we expect that $X/G$ has a variety structure (especially a scheme structure) it should be possible to talk about local sections of $X/G$. What are they?
If we can solve this question then the next problem would be how the $G$- action is extended the the coherent sheaves $f_*F$ as given in the exact sequence after applying $p_*$? or more generally any coherent sheaves $E$ on $S$?