Finite groups in which Sylow normalizers have nilpotent Hall supplements

The normalizer of each Sylow subgroup of a finite group $G$ has a nilpotent Hall supplement in $G$ if and only if $G$ is soluble and every tri-primary Hall subgroup $H$ (if exists) of $G$ satisfies either of the following two statements: (i) $H$ has a nilpotent bi-primary Hall subgroup; (ii) Let $\pi(H)=\{p,q,r\}$. Then there exist Sylow $p$-, $q$-, $r$-subgroups $H_p$, $H_q$ and $H_r$ of $H$ such that $H_q\subseteq N_H(H_p)$, $H_r\subseteq N_H(H_q)$ and $H_p\subseteq N_H(H_r)$.