Let $X$ be an algebraic stack and let $f: S \to X$ be a smooth covering of $X$ by a scheme $S$.
Motivation: Forgetting about stacks for a moment and going back to covering spaces: Given a covering map $f: Z \to Y$, with $Z$ connected and $Y$ locally connected, then $\operatorname{Aut}(Z/Y)$ acts properly and discontinuously on $Z$. Moreover, if $\operatorname{Aut}(Z/Y)$ acts transitively on a fiber of $p \in Y$, then the covering is a $G=\operatorname{Aut}(Z/Y)$-covering in the sense that $f: Z \to Y \cong Z/\operatorname{Aut}(Z/Y)$ is a quotient map.
I am interested in making an analogous statement in the case of a smooth cover $f:S \to X$ of an algebraic stack $X$. (Of course, dropping words like properly and discontinuously and keeping in mind that $f$ is not finite 'etale and thus not a covering map in the above sense).
In particular, I want to describe $\operatorname{Aut}(S/X)$. The "elements" of $\operatorname{Aut}(S/X)$ are maps $\phi: S \to S$ such that $f \circ \phi =f$. On the other hand, $f: S \to X$ can be identified with a unique object $s \in X(S)$ (up to $2$-isomorphism?) by the 2-Yoneda lemma and so it seems like $\operatorname{Aut}(S/X)$ should have an interpretation in terms of the groupoid of maps $s \to s$ lying over a given $\phi: S \to S$.
That is all very abstract, so let us just suppose that the elements of $X(S)$ have some geometric interpretation, for example, the object $s \in X(S)$ is a family of genus g curves $C$ over a scheme $S$.
(1). Is there an interpretation of the groupoid $s \to s$ lying over $\phi: S \to S$ in terms of a group automorphisms of $C$ over $S$?
(2). Moreover, how would this group act on a "fiber" $S \times_{X,g} T$ (a sheaf on the category $Sch/S \times T$?) over $g: T \to X$?
