A note on a result of Skiba

A subgroup $H$ of a group $G$ is called weakly $s$-permutable in $G$ if there is a subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\le H_{sG}$, where $H_{sG}$ is the maximal $s$-permutable subgroup of $G$ contained in $H$. We improve a nice result of Skiba to get the following
Theorem. Let $\mathscr F$ be a saturated formation containing the class of all supersoluble groups $\mathscr U$ and let $G$ be a group with $E$ a normal subgroup of $G$ such that $G/E\in\mathscr F$. Suppose that each noncyclic Sylow $p$-subgroup $P$ of $F^*(E)$ has a subgroup $D$ such that $1<|D|<|P|$ and all subgroups $H$ of $P$ with order $|H|=|D|$ are weakly $s$-permutable in $G$ for all $p\in\pi(F^*(E))$; moreover, we suppose that every cyclic subgroup of $P$ of order 4 is weakly $s$-permutable in $G$ if $P$ is a nonabelian 2-group and $|D|=2$. Then $G\in\mathscr F$.