On the formations of finite solvable groups with property ${\mathcal P}_{2}$

Given two classes ${\mathfrak F}$ and ${\mathfrak X}$ of finite groups, ${\mathfrak F}$ is said to have property ${\mathscr P}_{2}$ for ${\mathfrak X}$ whenever ${\mathfrak F}$ contains every ${\mathfrak X}$-group $G$ expressible as the product of some subgroups $A_{1}, A_{2}, \dots, A_{n}$ such that the groups $A_{i}A_{j}$ lie in ${\mathfrak F}$ for all $1\leq i<j\leq n$. This article describes all $Z$-saturated $s_F$-closed formations and Fischer formations of solvable groups with property ${\mathscr P}_2$. In particular, the set of all such formations coincides with the set of hereditary Shemetkov formations in the class ${\mathfrak S}$ of all finite solvable groups. We describe the hereditary saturated formations ${\mathfrak X}$ with every saturated subformation having property ${\mathscr P}_{2}$ for ${\mathfrak X}$.