Let $X$ be a smooth complex quasiprojective variety and let $G$ be a finite group acting on $X$. The quotient $X/G$ is then a variety. Let $\mathcal{F}$ be a $G$-equivariant coherent sheaf on $X$.
Question: Under what conditions will $\mathcal{F}$ descend to a coherent sheaf on $X/G$?
This will hold if $G$ acts freely (this can e.g. be found in Mumford's book on abelian varieties), but alas in the situation I am in this does not hold.
Dan Petersen's answer to this question says that this will hold if I am content to work in the category of stacks and use the stacky quotient, but I am not comfortable with stacks and would prefer to work in the world of varieties.