All varieties are assumed to be projective over $\mathbb{C}$. Let $f_1: Y \to X$ and $f_2: Y' \to X$ be étale morphisms with same finite Galois groups (to be honest, I don't know what does Galois group really mean in this context), then can we conclude $Y,Y'$ are birational?
Even if the general situation might have negative answer, are there any results concerning problemss along this line?