Products of $\mathrm{F}^*(G)$-subnormal subgroups of finite groups

A subgroup $H$ of a finite group $G$ is called $\mathrm{F}^*(G)$-subnormal if $H$ is subnormal in $H\mathrm{F}^*(G)$. We show that if a group $G$ is a product of two $\mathrm{F}^*(G)$-subnormal quasinilpotent subgroups, then $G$ is quasinilpotent. We also study groups $G=AB$, where $A$ is a nilpotent $\mathrm{F}^*(G)$-subnormal subgroup and $B$ is a $\mathrm{F}^*(G)$-subnormal supersoluble subgroup. Particularly, we show that such groups $G$ are soluble.