Finite and cyclic automorphism group implies cyclic

Statement
Suppose $$G$$ is a finite group that is an fact about::aut-cyclic group, i.e., the automorphism group of $$G$$ is a cyclic group. Then, $$G$$ is itself a fact about::cyclic group, and in particular, a fact about::finite cyclic group.

Note that the converse is not true, i.e., the automorphism group of a finite cyclic group need not be cyclic.

Similar facts

 * Trivial automorphism group implies trivial or cyclic of order two
 * Finite abelian and aut-abelian implies cyclic
 * Cyclic implies aut-abelian
 * Aut-cyclic implies abelian
 * Aut-abelian implies class at most two

Opposite facts

 * Aut-cyclic not implies cyclic
 * Aut-abelian not implies abelian
 * Abelian and aut-abelian not implies locally cyclic

Facts used

 * 1) uses::Aut-cyclic implies abelian
 * 2) uses::Cyclic implies abelian
 * 3) uses::Finite abelian and aut-abelian implies cyclic

Proof from given facts
Given: A finite group $$G$$ such that $$\operatorname{Aut}(G)$$ is cyclic.

To prove: $$G$$ is cyclic.

Proof