  L\'{a}szl\'{o} Fuchs posed the following question: which abelian groups arise
as the group of units in a ring? In this paper, we investigate a related
question: for such realizable groups $G$, when is there a ring $R$ with unit
group $G$ such that every group endomorphism of $G$ is induced by a ring
endomorphism of $R$? We answer this question for four common classes of groups:
torsion-free abelian groups, groups of odd order, torsion abelian groups, and
finitely generated abelian groups.
