A criterion for a finite group to belong a saturated formation

We prove the following result: Let $\mathcal{F}$ be a hereditary saturated formation of $p$-soluble groups containing all $p$-supersoluble groups such that $\mathcal{F}=\mathcal{G}_p\mathcal{F}$. Let $G=AT$ where $A$ is a Hall $\pi$-subgroup of $G$, $p\notin\pi$ and $T$ is a $p$-supersoluble subgroup of $G$. Suppose that for a Sylow $p$-subgroup $P$ of $T$ we have $|P|>p$. If $A$ permutes with a Hall $p'$-subgroup of $T$ and with all maximal subgroups $V$ of $P$ such that $G^{\mathcal{F}}\cap P\not\leqslant V$, then $G\in\mathcal{F}$.