I asked this question on Math Stack Exchange, but received no answer. I thought that maybe I would get an answer here.

Suppose that $X$ is a smooth $n$-dimensional quasiprojective variety over $\mathbb{C}$ and that $G$ is a finite group acting on $X$. Let $Y=X/G$. Let $\text{Pic}\ Y$ denote the Picard group of $Y$ and let $A_{n-1}(Y)$ denote the Chow group of codimension 1 cycles on $Y$ mod rational equivalence. Is it true that $\text{Pic}\ Y\otimes \mathbb{Q}\cong A_{n-1}(Y)\otimes \mathbb{Q}$?