Some permutability properties related to -hypercentrally embedded subgroups of finite groups

Let denote a saturated formation. In this paper we study some properties of -hypercentrally embedded subgroups, i.e., those subgroups T of a finite group G such that every chief factor of G between its core and its normal closure is -central in G. We prove that these subgroups form a sublattice of the lattice of all subgroups of G, if is subgroup-closed. The main result of the paper is the following: if contains the class of nilpotent groups and G is a soluble group, a subgroup T which permutes with all Sylow subgroups of G is -hypercentrally embedded in G if and only if T permutes with some -normalizer of G.