Suppose $X$ is a singular variety over a field $k$, which admits an action by a finite group $G$. Suppose the quotient $X/G$ is also a variety over $k$. If $Y_G$ is a resolution of $X/G$, does there exist a variety $Y$ which is a resolution of $X$ such that the action on $X$ could be extended to an action on $Y$ and

$$Y/G \simeq Y_G$$

Put it in another way, is the resolution of the quotient "the same thing" as the quotient of (equivariant) resolution?

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