Nilpotent not implies nilpotent automorphism group

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., nilpotent group) need not satisfy the second group property (i.e., aut-nilpotent group)
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Statement

The automorphism group of a nilpotent group need not be nilpotent. In other words, a nilpotent group is not necessarily an aut-nilpotent group (group having a nilpotent automorphism group).

Proof

Example of the Klein four-group

Further information: Klein four-group, symmetric group:S3

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.