TITLE: centre and automorphism groups of finite group schemes
QUESTION [5 upvotes]: Let $G$ be a group scheme over a scheme $X$ with centre $Z(G)$, automorphism group $\mathrm{Aut}(G)$ and outer automorphism group $\mathrm{Out}(G)$ (viewed as group schemes on $X$).


If $G$ is finite flat over $X$, then are $Z(G), \mathrm{Aut}(G)$ and $\mathrm{Out}(G)$ also finite flat over $X$?
If $G$ is finite étale over $X$, then are $Z(G), \mathrm{Aut}(G)$ and $\mathrm{Out}(G)$ also finite étale over $X$?

REPLY [2 votes]: The answer to 2. is yes.
A sketch of a proof is as follows: $G$, being finite étale, is étale locally on $X$ isomorphic to a constant finite group scheme. Therefore, by a standard descent argument, it suffices to prove the result for constant finite group schemes. However here $Z(G), \mathrm{Aut}(G)$ and $\mathrm{Out}(G)$ are all clearly finite étale, as required.