A note on $\sigma$-nilpotency of finite groups

Let $\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes, and $\sigma(n) =\{\sigma_i\mid i\in I, \sigma_i\cap\pi(n)\neq\emptyset\}$ for any integer $n$. A group $G$ is called $\sigma$-primary if either $G =1$ or $|\sigma(G)| =1$. A group $G$ is $\sigma$-nilpotent if $(H/K) \rtimes(G/C_G(H/K))$ is $\sigma$-primary for every chief factor $H/K$ of $G$. In this note, we prove that $G$ is $\sigma$-nilpotent if and only if $G$ is a $\sigma$-full group and $\pi(|xy|)=\pi(|x||y|)$ for any two elements $x,y\in G$ such that $\sigma(|x|)\cap\sigma(|y|)=\emptyset$.