Abelian automorphism group implies class two

Statement
Let $$G$$ be a group whose automorphism group is abelian, i.e., the automorphism group of $$G$$ is an abelian group. Then, $$G$$ is a group of nilpotency class two

Similar facts

 * Nilpotent automorphism group implies nilpotent of class at most one more
 * Trivial automorphism group implies trivial or order two

Other related facts

 * Cyclic implies abelian automorphism group
 * Abelian automorphism group not implies abelian
 * Locally cyclic implies abelian automorphism group
 * Abelian and abelian automorphism group not implies locally cyclic
 * Finite abelian and abelian automorphism group implies cyclic

Facts used

 * 1) uses::Group acts as automorphisms by conjugation
 * 2) uses::Abelianness is subgroup-closed

Proof
Since $$\operatorname{Aut}(G)$$ is an abelian group, the subgroup $$\operatorname{Inn}(G)$$, the inner automorphism group is also an abelian group. But $$\operatorname{Inn}(G) \cong G/Z(G)$$, where $$Z(G)$$ is the center of $$G$$. Thus, $$G/Z(G)$$ is abelian, so $$G$$ has class two.