On finite groups with generally subnormal sylow subgroups

Let $\mathfrak{F}$ be a non-empty formation. A subgroup $H$ of group $G$ is called $\mathfrak{F}$-subnormal in $G$ if either $H = G$ or there is a chain of subgroups $H = H_0 \subset H_1 \subset \dots \subset H_n = G$ such that $H_i^{\mathfrak{F}} \subseteq H_{i-1}$ for every $i = 1, \dots , n$. In the work the class of groups $w\mathfrak{F} = (G \mid\pi(G) \subseteq \pi(\mathfrak{F})$ and every Sylow subgroup of $G$ is $\mathfrak{F}$-subnormal in $G)$ are studied. Properties of the class $w\mathfrak{F}$ are obtained. In particular, for hereditary saturated formation $\mathfrak{F}$ it is proved that the class $w\mathfrak{F}$ is a hereditary saturated formation. Necessary and sufficient conditions are found, at which $w\mathfrak{F} = F$.