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.

Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. math. 97 (1989), no. 1, 53-94). The sheaf $\mathcal{F}$ descends if and only if for each point $x$ of $X$, the stabilizer of $x$ in $G$ acts trivially on the fiber $\mathcal{F}_x$. $\endgroup$