Group whose automorphism group is abelian

Symbol-free definition
A group is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if its defining ingredient::automorphism group is an defining ingredient::abelian group or equivalently, if any two automorphisms of the group commute.

Definition with symbols
A group $$G$$ is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if $$\operatorname{Aut}(G)$$ is an abelian group.

Formalisms
A group $$H$$ is a group whose automorphism group is abelian if and only if for every group $$G$$ containing $$H$$ as a normal subgroup, $$H$$ is also contained in the centralizer of derived subgroup of $$G$$.

Related subgroup properties

 * Aut-abelian normal subgroup is a normal subgroup of a group that is aut-abelian as a group.

Facts

 * Derived subgroup centralizes normal subgroup whose automorphism group is abelian: Any normal subgroup whose automorphism group is abelian commutes with every element in the derived subgroup. Hence, it is contained in the centralizer of commutator subgroup.
 * Finite abelian and abelian automorphism group implies cyclic