On Products of $\mathrm{F}^*(G)$-Subnormal Subgroups of Finite Groups

A subgroup $H$ of a finite group $G$ is said to be $\mathrm{F}^*(G)$-subnormal if it is subnormal in $H\mathrm{F}^*(G)$, where $\mathrm{F}^*(G)$ is the generalized Fitting subgroup of $G$. In the paper, the structure of groups factorizable by two $\mathrm{F}^*(G)$-subnormal subgroups, one of which is nilpotent, is studied. In particular, if the other factor is metanilpotent, then the group is solvable. Moreover, if the commutator subgroup of the second factor is nilpotent, then the nilpotent length of the group is at most 3. The supersolvability of the product $G=AB=AC=BC$ of a nilpotent subgroup $A$ by supersolvable subgroups $B$ and $C$ (all three are $\mathrm{F}^*(G)$-subnormal) is established.