On $\pi$-supersolvability of finite groups

A subgroup $H$ of a group $G$ is called $\mathbb{P}_{\pi}$-subnormal in $G$ if either $H=G$ or from $H$ to $G$ there exists a chain of subgroups, whose every index is either a prime in $\pi$ or a $\pi'$-number ($\pi$ is some set of primes). For a finite $\pi$-closed group with given $\mathbb{P}_{\pi}$-subnormal subgroups, the necessary and sufficient conditions of $\pi$-supersolvability are obtained.