If a morphism of curves induces Galois extension of function fields, does the galois group act transitively on the fibers?

Question

Let $f:X\to Y$ be a non-constant morphism of smooth projective curves defined over an algebraically closed field. Suppose that the corresponding field extension $K(X) \setminus K(Y)$ is Galois. Does $\operatorname{Gal}(K(X) \setminus K(Y))$ act transitively on the fibers of $f$ ? More precisely, if $f(P)=f(Q)$, then is there a $g \in \operatorname{Gal}(K(X) \setminus K(Y))$ such that the induced automorphism $\sigma_g$ of $X$ takes $P$ to $Q$ ?

I have tried thinking about it in various way but I have not been able to use the fact that the field extension is Galois. A hint would be greatly appreciated.