Let $X$ be a topological space equipped with an action of a group $G$. In the Tohoku paper, Grothendieck defined "$G$-sheaves" on $X$ as sheaves equipped with $G$-action on the etale space, compatible with the $G$-action on $X$. It is more or less the same as $G$-equivariant objects in the category of sheaves on $X$ with respect to $G$ acting on this category by endofunctors, but the notion of "$G$-equivariant sheaves" is taken over by sheaves equivariant with respect to a group scheme or to a continuous action of a topological group.
Suppose now that the action of $G$ on $X$ is free. Then the category of $G$-sheaves on $X$ is equivalent to the category of sheaves on $X/G$. I need this statement when $G$ acts on a manifold $X$ properly discontinuously, but I suppose this is true in all generality. I have a proof which works (at least for properly discontinuous actions), but it's a bit too long for such a classical-looking statement, so I am asking for a reference I can use for this. I spent quite a few hours googling and browsing the Stacks Project, without avail.
