Finite groups with strongly $\mathrm{K}$-$\mathfrak{F}$-subnormal Sylow subgroups

Let $\mathfrak{F}$ be a nonempty formation of groups. We call a subgroup $H$ of the group $G$ strongly $\mathrm{K}$-$\mathfrak{F}$-subnormal in $G$, if $N_G(H)$ is a $\mathfrak{F}$-subnormal subgroup in $G$. In this paper we study the question of belonging a finite group in which all Sylow subgroups of strongly $\mathrm{K}$-$\mathfrak{F}$ subnormal, to saturated formation $\mathfrak{F}$.