Groups whose non-permutable subgroups are metaquasihamiltonian

If X is a class of groups, define a sequence of group classes X-1, X-2, ..., X-k, ... by putting X-1 = X and choosing X-k(+)1 as the class of all groups whose non-permutable subgroups belong to X-k. In particular, if U is the class of abelian groups, U-2 is the class of quasimetahamiltonian groups, i.e. groups whose non-permutable subgroups are abelian. The aim of this paper is to study the structure of X-k-groups, with special emphasis on the case X = U. Among other results, it will also be proved that a group has a finite normal subgroup with quasihamiltonian quotient if and only if it is locally graded and belongs to U-k for some positive integer k.