On σ -permutably embedded subgroups of finite groups

Let σ = {σi: i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi: σi ∩ π(G)≠O}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σi-subgroup of G and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). G is said to be σ-full if G possesses a complete Hall σ-set. A subgroup H of G is σ-permutable in G if G possesses a complete Hall σ-set H such that HAx= AxH for all A ∈ H and all x ∈ G. A subgroup H of G is σ-permutably embedded in G if H is σ-full and for every σi ∈ σ(H), every Hall σi-subgroup of H is also a Hall σi-subgroup of some σ-permutable subgroup of G.