In Milne's Lectures of Etale Cohomology, page 27,the first line, he says(all schemes are supposed to be integral.):
Suppose $X_i$ and $X_j$ are finite etale over scheme X, then a homomorphism $\phi\in \hom_X(X_i,X_j)$ induces a homomorphism:$\phi':\operatorname{Aut}_{X}(X_i)\to \operatorname{Aut}_{X}(X_j) $.
How? I think we may need another morphism:$\theta:X_j\to X_i$, but can we build it from $\phi$?