  Let $\pi$ be a set of primes and $\mathfrak{F}$ be a formation. In this
article a properties of the class ${\rm w}^{*}_{\pi}\mathfrak{F}$ of all groups
$G$, such that $\pi(G)\subseteq \pi(\mathfrak{F})$ and the normalizers of all
Sylow $p$-subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$ for every
$p\in\pi\cap\pi(G)$ are investigated. It is established that ${\rm
w}^{*}_{\pi}\mathfrak{F}$ is a formation. Some hereditary saturated formations
$\mathfrak{F}$ for which ${\rm w}^{*}_{\pi}\mathfrak{F}=\mathfrak{F}$ are
founded.

Group Theory