2.23 (1966)

A subgroup $H$ of a group $G$ is called quasisubinvariant if there is a normal system of $G$ passing through $H$. Let $\mathfrak{K}$ be a class of groups that is closed with respect to taking homomorphic images. A group $G$ is called an $R^0(\mathfrak{K})$-group if each of its non-trivial homomorphic images has a non-trivial quasisubinvariant $\mathfrak{K}$-subgroup. Do $R^0(\mathfrak{K})$ and $\overline{RN}$ coincide when $\mathfrak{K}$ is the class of all abelian groups?