On normal complements of $\mathfrak {F}$-covering subgroups

If $\mathcal {F}$ is a suitably restricted formation, we show that an $\mathcal {F}$-covering subgroup $H$ which is a Hall subgroup of the finite, solvable group $G$ is complemented by the $\mathcal {F}$-residual of $G$, provided $H$ normalizes an $\mathcal {F}$-normalizer of $G$. In particular, $H$ is complemented by the $\mathcal {F}$-residual, if $H$ is an $\mathcal {F}$-normalizer of $G$. Further, if $\mathcal {F}$ is the class of nilpotent groups, then $H$ complements the nilpotent residual, if $G$ has pronormal system normalizers. Examples are given to show the necessity of the various hypotheses.