Weakly characteristic subgroups of finite groups

Suppose that $G$ is a finite group and $H\le K\le G$. We say that $H$ is called weakly characteristic in $K$ with respect to $G$ if for all subgroups $R$ of $K$ containing $H$, we have $H$ char $R$. In particular, when $H$ is a subgroup of prime power order and $K$ is a Sylow subgroup containing it, $H$ is simply said to be a weakly characteristic subgroup of $G$ or weakly characteristic in $G$. In this paper, we investigate the structure of finite groups by means of weakly characteristic subgroups.