Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$.
Are the following two conditions equivalent?
The function field extension $K(Y)\subset K(X)$ is Galois (in the field-theoretic sense)
The quotient $X/G$ exists and the natural morphism $X/G\to Y$ is an isomorphism.