A New Generalized $$p$$-Local Class of Groups $$\mathcal{S}^{p}$$ and Its Properties

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.