Class two not implies abelian automorphism group

Statement
A group of nilpotency class two need not be a group whose automorphism group is abelian, i.e., its automorphism group need not be an abelian group.

Related facts

 * Cyclic implies abelian automorphism group
 * Abelian automorphism group implies class two
 * Abelian automorphism group not implies abelian

Stronger facts

 * Weaker than::Finite abelian and abelian automorphism group implies cyclic

Example of the Klein four-group
Consider the Klein four-group: the direct product of two copies of the cyclic group of order two. It is an abelian group, hence it is a group of nilpotency class two.

On the other hand, its automorphism group is isomorphic to the symmetric group of degree three, which is not an abelian group.

Example of the dihedral group of order eight
The dihedral group of order eight is a non-abelian group of nilpotency class two. Its automorphism group is also isomorphic to it, i.e., it is a dihedral group of order eight. Thus, we have a group of class two whose automorphism group is not abelian.