centre and automorphism groups of finite group schemes 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 $X$).

  
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*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 $X$?
  

 A: The answers to 1. are all no since a finite flat group scheme can have fibers of very different isomorphism type. 
Here is an example of a finite flat group scheme where $Z(G)$ is not finite flat: Let $k$ be algebraically closed of odd characteristic $p$ and $X=\mathbf A^1$. Let $G$ be the closed subscheme of $GL_{3,X}$ given by matrices of the form
$$
A=\pmatrix{a&b&x\\sb&a&y\\0&0&1}
$$
subject to the relations $\det A=a^2-sb^2=1$ and $a^p-1=b^p=x^p=y^p=0$.
Here $s$ is the parameter on $X$. For $s\ne0$, the fiber of $G$ is the semidirect product $\mu_p\ltimes\alpha_p^2$ with $\mu_p$ acting by non-trivial characters. In particular, the center is trivial. For $s=0$ the group scheme is unipotent and its center is nontrivial.
A: 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.
