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 $G=Aut(k(X)/k(Y))$.

> 1. **When is there a well defined action of $G$ on $X$?** 
> 
> 2. **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 (2) always holds (although $G$ may be different) but I haven't been able to formalize this. Are there any useful conditions which ensure that (2) holds?