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) aS dagger N (H) (H (p) ), H (r) aS dagger N (H) (H (q) ), and H (p) aS dagger N (H) (H (r) ).