If $G$ is an affine group scheme over a field $k$, then we have the forgetful functor $\omega$ from the category ${\rm Rep}_k(G)$ of finite representations of $G$ to the category of finite dimensional $k$-vector spaces. Let $S$ be a $k$-scheme, with the structure map $p: S\to k$. Then ${\rm Aut}_S(\omega)$ (the tensor automorphisms the non-neutral fibre functor $p^*\omega$) is an $S$-group scheme. In fact it is the diagonal of the groupoid scheme ${\rm Aut}_k(\omega)$ (see Deligne's article "catégories tannakiennes" in The Grothendieck Festschrift Volume II, 1.11 and 1.12 for notations and statements).

My question is what is the relation between the $S$-grp sch ${\rm Aut}_S(\omega)$ and the $S$-grp sch $G\times_kS$? are they equal?