Coherent $G$-sheaf on algebraic varieties Let $X$ be an algebraic variety (i.e. an integral separated scheme of finite type over an algebraically closed field $k$) and let $G$ be a finite group of automorphisms of $X$. Suppose (as we may in the case of quasi-projective varieties) that for any $x$ in $X$, the orbit $G_x$ of $x$ is contained in an affine open subset of $X$. A classical result states that there exists a (unique) $k$-variety $Y$ together with a finite, surjective and separable morphism $\pi \colon X\to Y$ such that:
1) As topological space $(Y,\pi)$ is the quotient of $X$ for the action of $G$.
2)There is a natural isomorphism $\mathcal{O}_Y \to \pi_{*}(\mathcal{O}_X)^{G}$.
In this setting, let $\mathcal{F}$ be a coherent sheaf on $Y$. Since for any $g$ in $G$ we have a commutative diagram:
\begin{array}{ccc}
X & \to^{g} & X \\
\downarrow^{\pi} & &  \downarrow^{\pi}\\
Y & \to^{id} & Y 
\end{array}
there should be a map between $\pi^{*}\mathcal{F}$ and $\pi^{*}\mathcal{F}$ induced by $g$, that is a natural automorphism. However, I can't really understand what the map is supposed to do...
The conclusion is that $G$ acts on the pullback of the sheaf $\mathcal{F}$ in a compatible manner (with respect to the action on $X$), but I can hardly imagine what is really happening here, beyond the formal arguments. Can anybody explain this in a more concrete way?
P.S. Sorry for the bad $\TeX$ typesetting: I suppose that the above diagram should be understood as a commutative triangle.
 A: The slogan here is that questions about the (non-equivariant) geometry of $Y$ have answers coming from the $G$-equivariant geometry of $X$.
It's probably helpful to consider first the case of a topological space $X$ with a free action of a finite group $G$, and a vector bundle $E$ on $Y=X/G$. In this case, the pullback is by definition the vector bundle of pairs $(x,e)$ where $x \in X$ and $e$ is in the fiber of $e$ over the orbit of $x$. The group $G$ evidently acts on $X \times E$ via its action on $X$, and this set of pairs is stable. 
Now, to motivate the definition of a $G$-equivariant coherent sheaf, let's begin with a fiber bundle $E$ on a topological space $X$ with a continuous action of a topogical group $G$. There are two maps $\pi,a:G \times X \rightarrow X$, the projection $\pi$ onto the second factor and the action map $a$. We can pull $E$ back by these to obtain two bundles $\pi^*E$ and $a^*E$ on $G\times X$. Concretely, we have
$$\pi^* E=\{(g,x,e) \ | \ e \in \text{fiber through} \ x \} \quad \text{and} \quad
a^* E=\{(g,x,e) \ | \ e \in \text{fiber through} \ gx \},$$ and the map $\alpha:(g,x,e) \mapsto (g,x,ge)$ defines an isomorphism from $\pi^* E$ onto $a^*E$. The associativity for the action of $G$ on $E$ and the fact that $1 \in G$ acts by the identity force this map $\alpha$ to satisfy some additional requirements, which I will be happy to tell you about if this is what you want to know. 
This is the official definition of $G$-equivariant sheaf on $X$, which works in any geometric category (e.g., schemes, or algebraic spaces): it is a sheaf on $X$ together with an isomorphism $\alpha:\pi^* E \rightarrow a^* E$ satisfying additional conditions corresponding to $g(h(x))=(gh)(x)$ and $1 x=x$.
At the level of sections, given a $G$-equivariant sheaf $E$ and a section $f$ of $E$ over an open subset $U$ of $X$, we obtain a section $gf$ of $E$ over $gU$. But of course we want this to vary continuously in $g$, $f$, and $U$, and the definition we have indicated above packages all this conveniently.
