Existence of Hall normal subgroups in finite groups

We obtain a necessary and sufficient criterion for the existence of an invariant complement to a nilpotent subgroup contained as a direct factor in one of the maximal subgroups of a given group; we also find a condition for the $p$-closure of a group, all proper subgroups of which are $p$-closed, expressed in terms of the degree of one of its nonlinear irreducible characters.