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.Comment: 13 pages, Accepted for publication in the Journal of Algebr