Factorizations of One-Generated Composition Formations

A non-empty formation $\mathfrak F$ of finite groups is said to be solubly saturated, or we call it a composition formation, if every finite group $G$ having a normal subgroup $N$ such that $G/\Phi(N)\in\mathfrak F$ belongs to $\mathfrak F$. An intersection of all composition formations containing a given group $G$ is denoted $c\operatorname{form}G$. Conditions are described under which $\mathfrak F=c\operatorname{form}G$ has the form $\mathfrak F=\mathfrak{MH}$, where $\mathfrak M\ne\mathfrak F\ne\mathfrak H$.