Finite group with cyclic quotient of automorphism group by class-preserving automorphism group

Definition
A finite group with cyclic quotient of automorphism group by class-preserving automorphism group is a finite group that is also a group with cyclic quotient of automorphism group by class-preserving automorphism group, i.e., a group such that the quotient group of its automorphism group by the normal subgroup comprising the defining ingredient::class-preserving automorphisms is a cyclic group (and in particular, a finite cyclic group).