Nilpotent not implies nilpotent automorphism group

Statement
The automorphism group of a nilpotent group need not be nilpotent. In other words, a nilpotent group is not necessarily an group whose automorphism group is nilpotent.

Example of the Klein four-group
The automorphism group of the Klein four-group, which is an abelian and hence nilpotent group, is the symmetric group of degree three, which is not a nilpotent group.