Nilpotent automorphism group not implies abelian automorphism group

Statement
A group whose automorphism group is nilpotent (i.e., a group with nilpotent automorphism group) need not be a group whose automorphism group is abelian (i.e., a group with abelian automorphism group). In other words, the automorphism group of a group may be nilpotent and non-abelian.

Example of the dihedral group of order eight
The dihedral group of degree four and order eight, has automorphism group isomorphic to itself. The automorphism group is thus nilpotent but not abelian.