Finite Groups with Formation Subnormal Normalizers of Sylow Subgroups

Let $\mathfrak{F}$ be a formation. Properties of the class $\mathrm{w}^{*}\mathfrak{F}$ of all groups $G$ for which $\pi(G)\subseteq\pi(\mathfrak{F})$ and the normalizers of all Sylow subgroups are $\mathfrak{F}$-subnormal in $G$ are studied. In particular, it is established that this class is a formation closed with respect to taking Hall subgroups. Hereditary saturated formations $\mathfrak{F}$ coinciding with $\mathrm{w}^{*}\mathfrak{F}$ are found.