Let $S$ be a connected scheme, and let $f\colon X\to S$ be a finite étale cover. Then there exists a finite étale cover $Y\to S$ such that $X\times_S Y\cong \coprod_{i\in I} Y$.
Replacing $Y$ by one of its connected components, we may assume that $Y$ is connected. Then it is easy to see that the action of $\operatorname{Aut}_S(X)$ on $X\times_S Y$ is just an action on the index set $I$.
I am wondering, if the same is still true in case $Y$ is not necessarily connected. It seems to me that an automorphism of $X$ over $S$ might map two connected components of a copy of $Y$ in $X\times_S Y$ into two different copies.