Criterion for $\pi$-supersolvability for finite groups

It is proved that the class of finite $\pi$-supersolvable groups is precisely the class of all finite $\pi$-solvable groups with the following property: For each maximal subgroup $M$ of a $\pi$-solvable group $G$ with index $p^{\alpha}$ for some $p\in\pi$, there exists a cyclic subgroup $S$ of order $p^{\beta}(\beta\geqslant\alpha)$ such that $G=MS$ and $S$ commutes with each element of the Sylow system $\Sigma_M$ of the subgroup $M$.