Groups with normality conditions for non-abelian subgroups

Two relevant theorems of B.H. Neumann characterize groups with finite conjugacy classes of subgroups and groups in which every subgroup has finite index in its normal closure as central-by-finite groups and finite-by-abelian groups, respectively. These results have later been extended to the case of groups with similar restrictions on abelian subgroups. Moreover, Romalis and Sesekin have studied groups in which all non-abelian subgroups are normal, and in this paper we consider groups with normality conditions of Neumannʹs type for non-abelian subgroups.