Finite groups with subnormal residuals of Sylow normalizers

Considering a nonempty formation $\mathfrak{X}$ of nilpotent groups, we prove that a group $G$ is an extension of a nilpotent group by an $\mathfrak{X}$-group if and only if every Sylow normalizer in $G$ is solvable and its $\mathfrak{X}$-residual is subnormal in $G$. We also show that $G$ is supersolvable if and only if every Sylow normalizer in $G$ is supersolvable and its nilpotent residual is subnormal in $G$.