Groups in which every subgroup is permutable-by-finite

In two papers J.T. Buckley, J.C. Lennox, H. Smith, B.H. Neumann and J. Wiegold studied groups in which every subgroup contains a G-invariant subgroup of finite index, and, in particular, they proved that a generalized soluble group G for which there exists a positive integer n such that every subgroup contains a G-invariant subgroup of index at most n is abelian-by-finite. In this paper it is proved that a periodic generalized soluble group for which there exists a positive integer n such that every subgroup contains a permutable subgroup of G of index at most n is quasihamiltonian-by-finite. Moreover, some structural properties of non-periodic groups in which every subgroup contains a permutable subgroup of G of finite index are determined.