Let $G$ be a group acting faithfully and algebraically on an algebraic variety $X$. Then we can understand $G$ as a subgroup of the automorphisms group $Aut(X)$ of $X$. My question is: when is $G$ a normal subgroup of $Aut(X)$?
For instance, let us suppose that there exists a Galois cover $f\colon X\to Y$ that exhibits $Y$ as the quotient of $X$ by $G$. Is $G$ a normal subgroup of $Aut(X)$?
I found some references related to this in a continuous/differentiable setting, but I am struggling to find references in an algebraic setting. What are some reference I could take a look at?