Tannakian fundamental group for finitely linear representation of group  Let $G$ be an arbitrary group and $k$ a field. Denote by $Rep_{k}(G)$ the category of finite dimensional representations of $G$ over $k$. The usual tensor product and dual operations for representations equip $Rep_{k}(G)$ with the structure of a $k$-linear rigid abelian tensor category. The forgetful functor $Rep_{k}(G) \longrightarrow Vec_{k}$ as fiber functor. So we get a neutral Tannakian category.
My question is :
If we consider the $G$ as constant affine group scheme over $k$, does the Tannakian fundamental group isomorphic to $G$? 
 A: 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.
