I wonder for an explicit description of the "algebraic hull" (and it's associated Hopf algebra) of a given (discrete) group. I know the answer only for finite groups $G$ which is equal to Spec of the Cartier dual of $k[G]$. What's the answer for the simplest infinite group: $\mathbb{Z}$?.
(Algebraic hull: Tannakian fundamental group of $\mathrm{Rep}_k(G)$.)
A representation of $\mathbb{Z}$ is just a finite dimensional vector space over $k$ + an automorphism. The answer is probably quite complicated. For example, a representation of the one-dimensional Lie algebra $k$ is a finite dimensional vector space + an endomorphism. When $k$ is algebraically closed, the Tannakian fundamental group here is known, but it is complicated, namely, it is $Speck[t]×Speck[k]$. See question 21415 (answers of Milne/Ekedahl).
The 1954 paper of Iwahori also considers automorphisms, so it may be possible to deduce the group from his results, as in the case of an endomorphism.
If $k$ is an algebraically closed field of characteristic zero, Theorem 7 of Iwahori's 1954 paper "On some matrix operators" (J. Math. Soc. Japan vol. 6 no. 1, 76--105) parametrizes the set of p-p operators, also known as $k$-endomorphisms of the fiber functor $\omega$, by pairs $(g,d)$. Here, $g$ is an endomorphism of the multiplicative monoid $(k, \times)$, which acts on the eigenvalues of semisimple parts, and $d$ is an element of $k$, which describes the exponent on unipotent parts.
A bit of examination of Iwahori's argument shows that when $R$ is a $k$-algebra, the $R$-endomorphisms of the fiber functor are parametrized by pairs $(g,d)$ where $g: (k,\times) \to (R,\times)$ is a monoid homomorphism, and $d \in R$ is an element. These sets have natural monoid structures given by pointwise multiplication. We need an affine monoid scheme whose $R$-points satisfy this description.
The $k$-algebra maps from the monoid ring $k[(k,\times) \times (\mathbb{N},+)]$ to $k$ are in natural bijection with pairs $(g,d)$ given by a multiplicative monoid endomorphism $g: k \to k$ together with an element $d$ of $k$. That is, $k$-points of the spectrum of $k[(k,\times) \times (\mathbb{N},+)]$ are in bijection with p-p operators, and more generally, $R$-points of $R[(k,\times) \times (\mathbb{N},+)]$ are in bijection with pairs $(g,d)$ where $g$ is a homomorphism from $(k,\times)$ to $(R,\times)$, and $d$ is an element of $R$. The usual monoid ring coproduct $a \mapsto a \otimes a$, that makes basis elements grouplike, induces a pointwise multiplication operation on $R$-points, for which the constant map to $1$ is the identity. This is, therefore, a description of the bialgebra whose spectrum gives endomorphisms of the fiber functor.
The submonoid scheme of invertible elements is the algebraic hull of $\mathbb{Z}$ over $k$. Concretely, it is the spectrum of the group ring $k[k^\times \times \mathbb{Z}]$.
