Let $X$ be a scheme and let $G$ be an abstract group acting on $X$ by scheme automorphisms. I'm happy to assume finiteness conditions on $X$ (such as locally Noetherian) and on $G$ (such as $G$ is finite) as necessary.

I understand that it is possible to enlarge the category of schemes in such a way that there is a good "quotient space" $X/G$ associated to this data.

A. What is the correct framework in which to study $X/G$? Is this an example of a *quotient stack* or perhaps an *algebraic space*?

There should be a good category of sheaves on $X/G$. Let's agree that the "right" category of sheaves on the scheme $X$ is the category of quasi-coherent sheaves of $\mathcal{O}_X$-modules.

B. What is the "right" category of sheaves on $X/G$?

This should be an abelian category with enough injectives. Assuming the answer to this question is "yes", let's call this category $Qcoh(X/G)$.

C. Is it possible to realise $Qcoh(X/G)$ as a subcategory of the category of abelian sheaves $Sh(\mathcal{C})$ on some Grothendieck site $\mathcal{C}$?

One candidate for such a $\mathcal{C}$ is the set of $G$-stable open subsets of $X$, but presumably this is too naive to work in general.

D. Is there always a morphism of abelian categories $Qcoh(X) \to Qcoh(X/G)$?

On the other hand, it is possible to form the category of $G$-equivariant quasi-coherent sheaves $G-Qcoh(X)$ on $X$.

E. What, if any, is the connection between $G-Qcoh(X)$ and $Qcoh(X/G)$?

I suspect that the answer to this last question is "$Qcoh(X/G)$ is equal to $G-Qcoh(X)$ by definition", in which case the question becomes "Can you shed some light on this definition"?

Finally, references to the literature where similar questions are studied would be greatly appreciated.