On finite semi-$p$-decomposable groups

A finite group $G$ is called $p$-decomposable if $G=O_{p'}(G)\times O_p(G)$. We say that a finite group $G$ is semi-$p$-decomposable if the normalizer of every non-normal $p$-decomposable subgroup of $G$ is $p$-decomposable. We prove the following Theorem. Suppose that a finite group $G$ is semi-$p$-decomposable. If a Sylow $p$-subgroup $P$ of $G$ is not normal in $G$, then the following conditions hold: (i) $G$ is $p$-soluble and $G$ has a normal Hall $p'$-subgroup $H$. (ii) $G/F(G)$ is $p$-decomposable. (iii) $O_{p'}(G)\times O_p(G)=H\times Z_\infty(G)$ is a maximal $p$-decomposable subgroup of $G$, and $G/H\times Z_\infty(G)$ is abelian.