Let $k$ be an arbitrary field and $X$ be a proper scheme over $k$. Mastumura and Oort proved that the functor $S \longmapsto Aut_{S-sch}(X \times S)$ is representable by a group scheme, locally of finite type, which I denote by $\underline{Aut}(X)$. I am interested in the forms of such a group scheme. More precisely, if $X'$ is a form of $X$ (that is, a (proper) scheme such that $X'_{\overline{k}} \simeq X'_{\overline{k}}$, where $\overline{k}$ is an algebraic closure of $k$ and $X_{\overline{k}}$ stands for the base change $X \times \mathrm{Spec}(\overline{k})$) then by functoriality $\underline{Aut}(X')_{\overline{k}} \simeq \underline{Aut}(X'_{\overline{k}}) \simeq \underline{Aut}(X_{\overline{k}}) \simeq \underline{Aut}(X)_{\overline{k}}$, so that $\underline{Aut}(X')$ is a form of $\underline{Aut}(X)$.

Question 1: Let $G$ be a form of $\underline{Aut}(X)$. Does there exist a form $X'$ of $X$ such that $G \simeq \underline{Aut}(X')$?

Moreover, if $Z$ is a closed subscheme of $X$ then the functor $$S \longmapsto \{g \in \underline{Aut}(X)(S) \mid \text{the automorphism } g \text{ of } X \times S \text{ induces an automorphism of } Z \times S \}$$ is representable by a subgroup scheme of $\underline{Aut}(X)$, which I denote by $\underline{Aut}(X,Z)$ (see Demazure-Gabriel, Théorème II 1.3.6 p165). If $(X',Z')$ is a form of $(X,Z)$ (that is, $Z'$ is a closed subscheme of $X'$ and the isomorphism $X'_{\overline{k}} \simeq X_{\overline{k}}$ induces an isomorphism $Z'_{\overline{k}} \simeq Z_{\overline{k}}$) then as above $\underline{Aut}(X',Z')$ is a form of $\underline{Aut}(X,Z)$.

Question 2: Let $G$ be a form of $\underline{Aut}(X,Z)$. Does there exist a form $(X',Z')$ of $(X,Z)$ such that $G \simeq \underline{Aut}(X',Z')$?

I ask these questions in a quite broad generality. However, in the example I have in mind, $X$ and $Z$ are smooth, projective and geometrically integral, and $\underline{Aut}(X,Z)$ is a smooth connected affine group scheme. I would be happy if the answers were positive with these restrictions.

I suspect that this could be related to cohomology, but **I do not want to assume the field $k$ to be perfect** so the forms could be trivialised after an inseparable extension.