Finite group whose outer automorphism group is cyclic

Definition
A finite group whose outer automorphism group is cyclic is a finite group that is also a group whose outer automorphism group is cyclic, i.e., the outer automorphism group is a cyclic group (and in particular a finite cyclic group).

Facts

 * Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group