Following Niels's suggestion, I'm turning my comments into an answer.
The first point I want to make is the observation that if $G$ is an infinite group, then the constant group scheme $G_k$ is not affine over $k$. As a scheme, it is an infinite disjoint union of copies of the spectrum of $k$, and the ring of global functions is an infinite product of copies of $k$. If you take the spectrum of that ring, the underlying topological space is not your original discrete space, first of all because it is quasi-compact. I think it has strictly larger cardinality, due to the existence of lots of exotic prime ideals. I imagine there is some relation with Stone-Cech compactification, but I am too lazy to look it up or try to figure it out.
In general the group of automorphisms of the fiber functor is an object called the pro-algebraic hull of $G$. It is an initial object among all homomorphisms from $G$ to pro-algebraic groups, and exists by the completeness of the category of affine group schemes. There is a more concrete description given in the comments by ChrisLazda, and as you can see from his/her example, it can depend strongly on the base field.
As far as I can tell, the notion of pro-algebraic hull was folklore (or perhaps an exercise for graduate students) for a while. Googling yields a description in this paper, together with an application.