Criteria of $p$-supersolubility and supersolubility of finite groups

Let $G$ be a finite group, $A$ a subgroup of $G$. Then we say that $A$ is generalized quasinormal in $G$ if $A$ either covers or avoids every maximal pair $(K,H)$ of $G$. We say that $A$ is $m$-supplemented in $G$ if $G$ has a subgroup $T$ and a generalized quasinormal subgroup $C$ such that $G = AT$ and $T \cap A \le C \le A$. Based on these concepts new characterizations of finite $p$-supersoluble and supersoluble groups are obtained.