More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to the category of vector spaces over $k$. With the usual notation in the theory of Tannakian categories, is the natural map $G\longrightarrow \underline{\text{Aut}}^{\otimes} (\omega_0)$ an isomorphism of group schemes over $k$?

Yes, see Deligne–Milne, "Tannakian Categories", Proposition 2.8, which says precisely this. (It doesn't matter what kind of field $k$ is, but many of the more specific recognition theorems, such as that for reductivity, do.) I feel like I must be misunderstanding, since I understood that to be the theory of Tannakian categories at its most basic. 

