On finite groups similar to supersoluble groups

A subgroup $H$ of $G$ is called $\mathbf{P}$-subnormal in $G$ if either $H = G$ or there is a chain $H = H_0 \subset H_1 \subset \dots \subset H_{n-1} \subset H_n = G$ such that $|H_{i+1} : H_i |$ is a prime number for every $i = 0, 1, \dots , n-1$. For the set of $\pi$ primes the properties of $\mathrm w_\pi$-supersoluble groups $G$, i.e. groups for which for every $p \in \pi$ Sylow $p$-subgroup is $\mathbf{P}$-subnormal in $G$ are investigated. It is proved that the class of all $\mathrm w_\pi$-supersoluble groups is a normally hereditary formation, and the class of all soluble $\mathrm w_\pi$-supersoluble groups is a hereditary saturated formation. The properties of the groups, which are the product of $\mathbf{P}$-subnormal subgroups are obtained.