  Let $\sigma =\{\sigma_{i} | i\in I\}$ 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.
