Quotients and inclusions of finite quasiprimitive permutation groups

A permutation group is said to be quasiprimitive if each non-trivial normal subgroup is transitive. Finite quasiprimitive permutation groups may be classified into eight types, in a similar fashion to the case division of finite primitive permutation groups provided by the OʹNan–Scott Theorem. The action induced by an imprimitive quasiprimitive permutation group on a non-trivial block system is faithful and quasiprimitive, but may have a different quasiprimitive type from that of the original permutation action. All possibilities for such differences are determined. Suppose that G