On $MP$-closed saturated formations of finite groups

A class of groups $\mathfrak{F}$ is called $MP$-closed, if it contains every group $G=AB$ such that $\mathfrak{F}$-subgroup $A$ permutes with every subgroup of $B$ and $\mathfrak{F}$-subgroup $B$ permutes with every subgroup of $A$. We prove that the formation $\mathfrak{F}$ containing the class of all supersoluble groups is $MP$-closed if and only if the formation $F(p)$ is $MP$-closed for all prime $p$, where $F$ is maximal integrated local screen of $\mathfrak{F}$. In particular, we prove that the formation of all groups with supersoluble Schmidt subgroups is $MP$-closed.