Finite groups with HL-embedded subgroups

Let G be a finite soluble group and let F be a class of groups. A chief factor H/K of G is said to be F-central (in G) if (H/K) (sic) (G/C-G(H/K)) is an element of F; we write L-cF(G) to denote the set of all subgroups A of G such that every chief factor H/K of G between A(G) and A(G) is F-central in G. Let X be a set of subgroups of G. We say that a subgroup A of G is H-L-embedded in G provided A is a Hall subgroup of some subgroup E is an element of L. In this paper, we study the structure of G under the condition that every subgroup of G is H-L-embedded in G, where L = L-cF(G) for some hereditary saturated formation F. Some known results are generalized.