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.

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