$\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups of finite groups

Let $\mathfrak F$ be a nonempty formation of groups, $\tau$ a subgroup functor, and $H$ a $p$-subgroup of a finite group $G$. Suppose also that $\bar G=G/H_G$ and $\bar H=H/H_G$. We say that $H$ is $\mathfrak F_\tau$-embedded ($\mathfrak F_{\tau,\Phi}$-embedded) in $G$ if, for some quasinormal subgroup $\bar T$ of $\bar G$ and some $\tau$-subgroup $\bar S$ of $\bar G$ contained in $\bar H$, the subgroup $\bar H\bar T$ is $S$-quasinormal in $\bar G$ and $\bar H\cap\bar T\le\bar SZ_\mathfrak F(\bar G)$ (resp., $\bar H\cap\bar T\le\bar SZ_{\mathfrak F,\Phi}(\bar G)$). Using the notions of $\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups, we give some characterizations of the structure of finite groups. A number of earlier concepts and related results are further developed and unified.