Finite groups whose n-maximal subgroups are σ-subnormal

Let σ = {σi | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hallσ-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 σi ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HAx = AxH for all A ∈ H and x ∈ G: σ-subnormal in G if there...