On the influence of the fitting subgroup on the products of finite soluble groups

A subgroup $H$ of a group $G$ is called $\mathrm{F}(G)$-subnormal if it is subnormal in $H\mathrm{F}(G)$. In the paper all closed under taking Schmidt subgroups saturated formations $\mathfrak{F}$ of finite soluble groups such that $\mathfrak{F}$ contains every soluble group $G$ which is the product of its $\mathrm{F}(G)$-subnormal $\mathfrak{F}$-subgroups were described. It was shown that a group $G$ which contains three supersoluble $\mathrm{F}(G)$-subnormal subgroups of pairwise coprime indexes in $G$ is supersoluble.