About the intersection of the maximal subgroups of finite groups

The structure of normal subgroups in $\Theta $-frattini expansions is established in the given work. Local Fitting $\frak F$ formation contains all nilpotent groups. For this formations we show that in solvable group the intersection $\frak F$-abnormal maximal $\Theta$-subgroups, which don't contain $\frak F$-radical and don't belong to $\frak F$, coincides with the intersection $\frak F$-abnormal maximal $\Theta $-subgroups and belongs formation $\frak F$.