A new generalized $p$-local class of groups $\mathcal{S}^{p}$ and its properties

A class of groups $\mathcal{S}^{p}$ containing every group $G$ whose any $pd$-chief factor $A/B$ of $G$ satisfies $| \Phi\bigl((A/B)_{p}\bigr)| \leqslant p$. We call a subgroup $H$ is a $\operatorname{CAP}_{\mathcal{S}^{p}}$-subgroup of a finite group $G$ if for any $pd$-chief factor $A/B$ of $G$, we have either $HA=HB$ or $| \Phi\bigl((H\cap A/H\cap B)_{p}\bigr)| \leqslant p$. Some characterizations for a finite group belongs to $\mathcal{S}^{p}$ are obtained under the assumption that some of its second maximal subgroups have generalized cover and avoidance properties.