On FΦ⁎-hypercentral subgroups of finite groups

Let G be a finite group. We write R(G) to denote the largest soluble normal subgroup of G and put Φ⁎(G)=Φ(R(G)). We say that a chief factor H/K of G is non-Frattini (non-solubly-Frattini) if H/K⩽̸Φ(G/K) (if H/K⩽̸Φ⁎(G/K), respectively). A chief factor H/K of G is called F-central in G provided (H/K)⋊(G/CG(H/K))∈F. A normal subgroup N of G is said to be FΦ-hypercentral (FΦ⁎-hypercentral) in G if either N=1 or N≠1 and there exists a chief series 1=N0