On $Phi$-$tau$-quasinormal subgroups of finite groups

‎Let $tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$‎. ‎Let $bar{G}=G/H_{G}$ and $bar{H}=H/H_{G}$‎. ‎We say that $H$ is $Phi$-$tau$-quasinormal in $G$ if for some $S$-quasinormal subgroup $bar{T}$ of $bar{G}$ and some $tau$-subgroup $bar{S}$ of $bar{G}$ contained in $bar{H}$‎, ‎$bar{H}bar{T}$ is $S$-quasinormal in $bar{G}$ and $bar{H}capbar{T}leq bar{S}Phi(bar{H})$‎. ‎In this paper‎, ‎we study the structure of a group $G$ under the condition that some primary subgroups of $G$ are $Phi$-$tau$-quasinormal in $G$‎. ‎Some new characterizations about $p$-nilpotency and solubility of finite groups are obtained.