On $\mathcal{U}\Phi$-hypercentre of finite groups

The product of all normal subgroups of $G$ whose all non-Frattini $G$-chief factors are cyclic is called the $\mathcal{U}\Phi$-hypercentre of $G$. The following theorem is proved.
Theorem. Let $X \le E$ be soluble normal subgroups of $G$. Suppose that every maximal subgroup of every Sylow subgroup of $X$ conditionally covers or avoids each maximal pair $(M,G)$, where $MX = G$. If $X$ is either $E$ or $F(E)$, then. $E \le Z_{\mathcal{U}\Phi}(G)$.