A constructive approach: From local subgroups to new classes of finite groups

Let G be a finite group and S be a proper subgroup of G. A group G is called an S-(CAP)-group if every local subgroup of G is either a CAP-subgroup or conjugate to a subgroup of S. The main purpose of this construction is to demonstrate a new way of analyzing the structure of a finite group by the properties and the number of conjugacy classes of its local subgroups.