On one generalization of finite nilpotent groups

Let $sigma ={sigma_{i} | iin I}$ be a partition of the set $Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $sigma$-central if the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is a $sigma_{i}$-group for some $i=i(H/K)$. $G$ is called $sigma$-nilpotent if every chief factor of $G$ is $sigma$-central. We say that $G$ is semi-${sigma}$-nilpotent (respectively weakly semi-${sigma}$-nilpotent) if the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal) $sigma$-nilpotent subgroup $A$ of $G$ is $sigma$-nilpotent. In this paper we determine the structure of finite semi-${sigma}$-nilpotent and weakly semi-${sigma}$-nilpotent groups.