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 $\dim X=\dim Y =1$?
I don't think that there is any (canonical) map $Aut(Y) \to Aut(X)$ of the kind you presume exists.
E.g. if $X$ is a curve of genus $g \geq 2$ and $Y$ is $\mathbb P^1$, then $Aut(X)$ is finite (often trivial), while $Aut(Y)$ equals $PGL_2(\mathbb C)$, which is simple. What is the map that you have in mind? (In any case, whatever it it is, it won't be injective.)