Let $f: X \to Y$ be a finite map of projective varieties.

I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Aut(X/Y)$.

1. When is $f$ isomorphic to the quotient map by an action $X \to X^G$?

If $X$ and $Y$ are smooth projective curves then I think this always holds but I haven't been able to formalize this. Are there any useful conditions which ensure that (1) holds?

EDIT: I deleted the irrelevant part of the question. I state it as a different question: When is $Aut(X/Y)=Aut(k(X)/k(Y))$?