On the XΦ-hypercentre of finite groups

Let G be a finite group, X a class of groups. A chief factor H/K of G is called X-central provided [H/K](G/CG(H/K))∈X. Let ZXΦ(G) be the product of all normal subgroups H of G such that all non-Frattini G-chief factors of H are X-central. Then we say that ZXΦ(G) is the XΦ-hypercentre of G. Our main result here is the following (Theorem 1.4): Let X⩽E be normal subgroups of a group G. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1<|D|<|P| and every subgroup H of P with order |H|=|D| and every cyclic subgroup of P with order 4 (if |D|=2 and P is a non-abelian 2-group) is weakly S-permutable in G. If X is either E or F∗(E), then E⩽ZUΦ(G). Here U is the class of all supersoluble finite groups.