On weakly m-σ-permutably embedded subgroups of finite groups

Let σ=\σᵢ|i∈ I\ be some partition of the set of all primes P and G a finite group. A group is said to be σ-primary if it is a finite σᵢ-group for some i.A subgroup H of G is said to be: σ-subnormal in G if there exists a subgroup chain H=H₀≤ H₁≤ ⋯ ≤ Hₙ=G such that either Hᵢ₋₁ is normal in Hᵢ or Hᵢ/(Hᵢ₋₁)Hᵢ is σ-primary for all i=1,…,n; σ-permutably embedded in G if H is σ-full and for every σᵢ∈ σ(H), every Hallσᵢ-subgroup of H is also a Hall σᵢ-subgroup of some σ-permutable subgroup of G.We say that a subgroup H of G is: m-σ-permutably embedded in G ifH= A, B for some modular subgroup A and σ-permutably embedded subgroup B of G;weakly m-σ-permutably embedded in G if there are an m-σ-permutably embedded subgroup L and aσ-subnormal subgroup T of G such that $G=HT$ and H∩ T≤ L ≤ H.