On Finite Groups with $p$-Decomposable Cofactors of Subgroups

It is proved that, for a finite group with $p$-decomposable cofactors of the maximal subgroups, the quotient group by the Fitting subgroup is $p$-decomposable. This implies that every group all of whose maximal subgroups have nilpotent cofactors is metanilpotent.