Here is a slightly anecdotical notational question.


Let $S$ be a scheme and let $X$ be a scheme over $S$, with structural morphism $s\colon X\to S$. Is there a good suggestive notation for the group $\lbrace (f,g)\in \mathrm{Aut}(X)\times \mathrm{Aut}(S)~\vert~sf=gs\rbrace $ ?


After chatting with a categorical friend, this can be described succinctly as the automorphism group of $X\to S$ (considered as an object in the arrow category of schemes).


In fact, I'm interested in finding a good notation when $S$ is (the spectrum of) a field $k$ and $X$ is an affine algebraic group $G$ over $k$. I thought about using $\mathrm{Aut}(G\to k)$ or $\mathrm{Aut}(G\to \mathrm{Spec}(k))$, but there might be something more adapted or already existing in the literature.



Notation for the automorphisms of a $S$-scheme over automorphisms of $S$
Indeed, this already exists in the literature. The automorphisms of $G\to\mathrm{Spec}(k)$ are called *semilinear automorphisms* of $G$, or just *semi-automorphisms* of $G$, and the corresponding group is denoted by $\mathrm{SAut}(G)$. See Subsection 3.2 of [this paper](https://arxiv.org/abs/1207.1329) and references therein.


