A note on finite groups with subnormal residuals of some Sylow normalizers

Let $G$ be a group and the set of primes $\tau(G)=\cup\pi(G : M)$ for any maximal subgroup $M$ of $G$. For a non-empty nilpotent formation $\mathfrak{X}$, it is proved that a group $G$ has a nilpotent $\mathfrak{X}$-residual if and only if the $\mathfrak{X}$-residual of the $p$-Sylow normalizer is subnormal in $G$ for every $p$ from $\tau(G)$.