On the generalized norm of a finite group

Let $G$ be a finite group and $\pi=\{p_1,\dots,p_n\}\subseteq\mathbb{P}$. Then $G$ is called $\pi$-special if $G=O_{p_1}(G)\times\dots\times O_{p_n}(G)\times O_{\pi'}(G)$. We use $\mathfrak{N}_{\pi sp}$ to denote the class of all finite $\pi$-special groups. Let $\mathrm{N}_{\pi sp}$ be the intersection of the normalizers of the $\pi$-special residuals of all subgroups of $G$, that is, $\mathrm{N}_{\pi sp}(G)=\bigcap\limits_{H\leqslant G}N_G(H^{\mathfrak{N}_{\pi sp}})$. We say that $\mathrm{N}_{\pi sp}$ is the $\pi$-special norm of $G$. We study the basic properties of the $\pi$-special norm of $G$. In particular, we prove that $\mathrm{N}_{\pi sp}$ is $\pi$-soluble.