Given a site $C$ and an object $U$, let $G$ be a sheaf of groups on this site and let $F$ be $G$-torsor, see the [Stacks Project](https://stacks.math.columbia.edu/tag/03AG) for the general definition.

By restriction on the under category $C/U$ (also a site with obvious topology), we get a $G|_U$-torsor $F|_U$ on $C/U$.

I want to first confirm that if it is true that the automorphism group of the trivial $G|_U$-torsor is identified with the group $G(U)$, the global section of the sheaf $G|_U$. And second, is there a nice description of the $G|_U$-torsor $F|_U$ on $C/U$ for a general $G$-torsor $F$?