On radicals of factorized finite $\mathfrak{X}$-groups

Let $\mathfrak{X}$ be a saturated $S$-closed formation of finite soluble groups containing the class $\mathfrak{N}^k$ of all groups whose nilpotent length does not exceed $k$, where $k\geqslant 3$. In this paper, we obtain a constructive description of all Fitting formations $\mathfrak{F}$ in $\mathfrak{X}$ such that for any $\mathfrak{X}$-group $G = AB$, there is $A_{\mathfrak{F}}\cap B_{\mathfrak{F}}\subseteq G_{\mathfrak{F}}$.