Given a finite group $G$, is $G$ determined by its category of maps from abelian groups? Specifically, we can form the category $G_A$ of "abelian points" of $G$ with objects pairs $(A,\phi)$, with $\phi:A\rightarrow G$, for $A$ abelian, with morphisms $\xi:(A,\phi)\rightarrow (B,\psi)$ given by morphisms of groups $\xi:B\rightarrow A$ such that $\phi \circ \xi = \psi$.
Very similar questions have been asked before, but they focused on the maps from cyclic groups, and the (small) counterexamples given for the analagous question do not have isomorphic categories of abelian points.
The following are the similar questions I refer to: Is a finite group uniquely determined by the orders of its elements? If I know the order of every element in a group, do I know the group?