On the strongly closed subgroups or $\mathscr H$-subgroups of finite groups

Let $G$ be a finite group. Goldschmidt, Flores, and Foote investigated the concept: Let $K\le G$. A subgroup $H$ of $K$ is called strongly closed in $K$ with respect to $G$ if $H^g\cap K\le H$ for all $g\in G$. 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 strongly closed subgroup. Bianchi and the others called a subgroup $H$ of $G$ an $\mathscr H$-subgroup if $N_G(H)\cap H^g\le H$ for all $g\in G$. In fact, an $\mathscr H$-subgroup of prime power order is the same as a strongly closed subgroup. We give the characterizations of finite non-$\mathscr T$-groups whose maximal subgroups of even order are solvable $\mathscr T$-groups by $\mathscr H$-subgroups or strongly closed subgroups. Moreover, the structure of finite non-$\mathscr T$-groups whose maximal subgroups of even order are solvable $\mathscr T$-groups may be difficult to give if we do not use normality.