So, I feel like I'm missing something obvious, but I have the following situation:

Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to Y$ be any resolution of singularities. Then we have a natural map $X\times_Y \tilde{Y}\to X$ which is birational, and $X\times_Y \tilde{Y}\to \tilde{Y}$ is a quotient by $G$. I want to be able to say that $X\times_Y \tilde{Y}$ is a resolution of $X$, and if $G$ acts freely, I can see how. But what if there's a locus (let's say a divisor) where $G$'s action has stabilizers?

So, I feel like I'm missing something obvious, but I have the following situation:

Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to Y$ be any resolution of singularities. Then we have a natural map $X\times_Y \tilde{Y}\to X$ which is birational, and $X\times_Y \tilde{Y}\to \tilde{Y}$ is a quotient by $G$. I want to be able to say that $X\times_Y \tilde{Y}$ is a resolution of $X$, and if $G$ acts freely, I can see how. But what if there's a locus (let's say a divisor) where $G$'s action has stabilizers?

EDIT: I'm leaving the above, though that situation is no good, thanks to comments below and abx's answer. But perhaps this is more reasonable:

Given $X\to Y$ a quotient by $G$ a finite group and $\tilde{Y}\to Y$ a resolution of $Y$, can I always find a resolution $\tilde{X}\to X$ such that $G$ acts on $\tilde{X}$ and $\tilde{X}/G\cong \tilde{Y}$?