A problem on Abelian groups

We solve Problem 44 in the book by L. Fuchs “Infinite Abelian Groups”, Vol. I, which asks for a classification of the groups $G$ having the following property: if $G$ is contained in the direct sum of reduced groups, then $nG$ for some $n>0$ is contained in a finite direct sum of these groups. A group has this property if and only if it has no unbounded factor groups that are direct sums of periodic cyclic groups. We also consider a generalization of this problem, when instead of the class of all reduced groups we take an arbitrary class of groups. We derive a number of properties of such groups.
Bibliography: 8 titles.