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 $G=Gal(\widetilde{k(X)}/k(Y))$ where $\widetilde{k(X)}$ is the Galois closure.
When is there a well defined action of $G$ on $X$? (Generically such an action always exists).
When is $f$ isomorphic to the quotient map by the Galois action $X \to X^G$?
If $X$ and $Y$ are smooth projective curves then I think this is always the case but I haven't been able to formalize this. Are there any useful conditions which ensure that (2) holds?