Abstract
The class of groups \(\mathcal{S}^{p}\) contains every group \(G\) such that any \(pd\) -chief factor \(A/B\) of \(G\) satisfies \(| \Phi\bigl((A/B)_{p}\bigr)| \leq p\) . We say that 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 \(| \Phi\bigl((H\cap A/H\cap B)_{p}\bigr)| \leq p\) . Some characterizations for a finite group to belong to \(\mathcal{S}^{p}\) are obtained under the assumption that some of its second maximal subgroups have generalized cover and avoidance properties.
