Let $X$ be an $S$-scheme and $G$ a finite group acting on $X$ over $S$. Denote the quotient stack $[X/G]$ by $\mathcal{X}$ and consider a schematic morphism of stacks $\mathcal{Y} \to \mathcal{X}$. Then the fiber product $X \times_{\mathcal{X}} \mathcal{Y}$ is representable by a scheme, say $Y$, on which $G$ acts in such a way that the first projection $Y \to X$ is $G$-equivariant.
My question is: under which conditions (if any) on the morphism $\mathcal{Y} \to \mathcal{X}$ is it true that the second projection $Y \to \mathcal{Y}$ induces an isomorphism between $[Y/G]$ and $\mathcal{Y}$?
P.S. I hope this is not a "stupid" question, but I'm not comfortable with stacks, so any help would be appreciated.