Finite groups with absolutely formationally subnormal Sylow subgroups

Let $\mathfrak{F}$ be a nonempty formation groups. A subgroup $H$ of a group $G$ we call absolutely $\mathrm{K}$-$\mathfrak{F}$-subnormal (absolutely $\mathfrak{F}$-subnormal) in $G$ if any subgroup containing $H$ is $\mathrm{K}$-$\mathfrak{F}$-subnormal (respectively $\mathfrak{F}$-subnormal) subgroup of $G$. In this paper, the question of whether a finite group belongs to a saturated formation $\mathfrak{F}$ in which all Sylow subgroups are absolutely $\mathrm{K}$-$\mathfrak{F}$-subnormal is investigated.