On finite semi-$\pi$-special groups

A finite group $G$ is called $\pi$-special if $G=O_{p_1}(G)\times\dots\times O_{p_n}(G)\times O_{\pi'}(G)$, where $\pi=\{p_1,\dots, p_n\}$. We say that a finite group $G$ is semi-$\pi$-special if the normalizer of every non-normal $\pi$-special subgroup of $G$ is $\pi$-special. We prove that if $G$ is not $\pi$-special but $N_G(A)$ is $\pi$-special for every subgroup $A$ of $G$ such that $A$ is either a $\pi'$-group or a $p$-group for some $p\in\pi$, then the following statements hold: (i) $G/F(G)$ is $\pi$-special. Hence $G$ has a Hall $\pi'$-subgroup $H$ and a soluble Hall $\pi$-subgroup $E$. (ii) If $G$ is not $p$-closed for each $p\in\pi$, then: (1) $H$ is normal in $G$ and $E$ is nilpotent. (2) $O_{p_1}(G)\times\dots\times O_{p_n}(G)\times H$ is a maximal $\pi$-special subgroup of $G$ and every minimal normal subgroup of $G$ is contained in $F(G)$.