What is a classical and perhaps even original reference for the following result, often called Tannaka reconstruction?
Let $G$ be a group and $R$ be a commutative ring in which $0,1$ are the only idempotent elements (for example, a field). Then the automorphism group of the tensor functor $U : \mathsf{Rep}_R(G) \to \mathsf{Mod}(R)$ (which forgets the $G$-action) is canonically isomorphic to $G$.
If $R$ is an arbitrary commutative ring, then the automorphism group is $C(\mathrm{Spec}(R),G)$.
There is the article of Deligne and Milne available here
http://www.jmilne.org/math/xnotes/tc.pdf
See Proposition 2.8. In the language you use, this is likely to be as classical as it gets.