Let $G$ be a discrete abelian group and $G^{\mathbb{N}}$ be the direct product (with the product topology), which consists of sequences $(a_1,a_2,a_3,\dots)$ of elements of $G$.

Let $G^{\infty}$ denote the group of sequences with values in $G^{\mathbb{N}}$, which converge to the identity $(0,0,0,\dots)\in G^{\mathbb{N}}$. Hence, $G^{\infty}$ is naturally a subgroup of $(G^{\mathbb{N}})^{\mathbb{N}}$. 

Is it possible to characterize the isomorphism type of $G^{\infty}$ in terms of the group $G$ or, perhaps, in terms of the structural characterization of infinite abelian groups, à la Fuchs, given a structural characterization of $G$? 

I am curious about the general problem but am most interested in the case where $G$ is finitely generated. Also, I am aware of several results that help identify the isomorphism type of various subgroups of the Baer-Specker group $\mathbb{Z}^{\mathbb{N}}$ but it's not so clear to me how this applies or if it informs this more general situation.