Suppose $X$ is a smooth variety and $G$ is a finite group acting on $X$. $X/G$ is not locally factorial.
Let $h: X\rightarrow X/G$ be the quotient morphism. Suppose there is a coherent sheaf $F'$ on $X/G$ which pullsback to $F$ on $X$. Since $X$ is smooth, one can define the determinant of $F$.
Can one define $det(F') \in Pic(X/G)$, such that $det(h^*F)=h^*(det F')$ ?
I looked into Mumford-Knudsen" preliminaries on Div and det", but could not understand if this is possible.