# Locally free resolution of sheaves on finite group quotient

Suppose $X$ is a smooth variety and consider a finite group $G$ acting on $X$. assume that the quotient map $X\rightarrow X/G$ is etale outside a codimension two subset. Suppose $H$ is coherent sheaf on $X/G$. Does there exist a finite locally free resolution for $H$ on $X/G$.

Thanks.

The answer is no as soon as $X/G$ is singular (which is quite often --- the simplest example is ${\mathbb C}^2/\{\pm 1\}$). You can take $H$ to be the structure sheaf of a singular point, then it does not have a finite locally free resolution.