Let $X$ and $Y$ be (smooth projective connected) varieties over $\mathbf{C}$.
Let $\pi:X\to Y$ be a finite surjective flat morphism.
Does this induce (by base change) a map $\mathrm{Aut}(Y) \to \mathrm{Aut}(X)$?
I think it does. Given an automorphism $\sigma:Y\to Y$, the base change via $\pi:X\to Y$ gives an automorphism of $X$.
My real question is as follows:
Is $\mathrm{Aut}(Y)\to \mathrm{Aut}(X)$ injective?
If not, under which hypotheses is $\mathrm{Aut}(Y)\to \mathrm{Aut}(X)$ injective? Does $\pi$ etale do the trick?
What if $\dim X=\dim Y =1$?