Finite groups with permutable Hall subgroups

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Let σ={σᵢ | i∈ I} be a partition of the set of all primes P and G a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σᵢ-subgroup of G for some i∈ I and H contains exactly one Hall σᵢ-subgroup of G for every i such that σᵢ∩ π(G)≠. In this paper, we study the structure of G assuming that some subgroups of G permutes with all members of H.