Group whose automorphism group is cyclic

Definition
An aut-cyclic group is a group whose automorphism group is a cyclic group.

Facts

 * Finite and aut-cyclic implies cyclic and classification of cyclic aut-cyclic groups
 * Aut-cyclic not implies cyclic

Stronger properties

 * Weaker than::Group of prime order: See multiplicative group of finite field is cyclic
 * Cyclic group of prime power order $$p^n$$, where $$p$$ is odd
 * Cyclic group:Z4.

Weaker properties

 * Stronger than::abelian group: See aut-cyclic implies abelian