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 (in nice cases) of the automorphism group of $G|_U$-torsor $F|_U$ on $C/U$ for a general $G$-torsor $F$? In my application, the group is highly nonabelian.

$\mathbf{Edit}:$ I think the answer to the first question is affirmative. For the second, I think in nice cases, $G(U)$ should be a subgroup of the automorphism group of $G|_U$-torsor $F|_U$. E.g. I think any $GL_n$-torsor (corresponds to a rank $n$ vector bundle on $U$) if the site is a category of schemes with e.g. Zariski topology should be a bitorsor (I hope this is correct)—it has actions by $GL_n$ from left and right, and the two actions commute, Then any element in $G(U)$ will give an automorphism of $F|_U$ by multiplying it on the right.

I hope some expert can give a definitive answer. Thanks in advance!