Let $X1$,$X2$ be two finite etale covering over $X$ with the same number of the fibers over $x_0$,which is a geometric point in $X$.And I assume that $X$ is a smooth projective variety over an algebraic closed field with character $0$.
My question is: is there an isomophism between $X1$ and $X2$ which commutes with the etale coverings？
Counterexample (characteristic different from $2$): a cyclic map of degree $4$ between elliptic curves $X_1 \to X$ on the one hand, and the multiplication by $2$ map $X_2=X \to X$ on the other. (Both are etale of degree $4$.)