On $\sigma_3$-nilpotent finite groups

Throughout the article all groups are finite and $G$ always denotes finite group; $\mathbb{P}$ is the set of all prime numbers and $\mathfrak{J}$ is some class of groups, closed under extensions, homomorphic images and subgroups. In this paper, $\sigma_3=\{\sigma_0\}\cup\{\sigma_i\mid i\in I\}$ is a partition of the set $\mathbb{P}$, i. e. $\mathbb{P}=\sigma_0\cup\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all indices $i\ne j$ from $\{0\}\cup I$, for which $\mathfrak{J}$ is a class of $\sigma_0$-groups with $\pi(\mathfrak{J})=\sigma_0$. The group $G$ is called: $\sigma_3$-primary if $G$ is either an $\mathfrak{J}$-group or a $\sigma_i$-group for some $i\ne0$; $\sigma_3$-nilpotent if $G$ is the direct product of some $\sigma_3$-primary groups. Finite $\sigma_3$-nilpotent groups are characterized.