Center not is 1-automorphism-invariant

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This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., 1-automorphism-invariant subgroup)
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Statement

The center of a group need not be a 1-automorphism-invariant subgroup.

Related facts

Proof

Let p be an odd prime, let G be the prime-cube order group:U(3,p), i.e., the unique non-abelian group of order p^3 and exponent p, and let H be the center of G. H is a cyclic subgroup of order p in G.

There exist 1-automorphisms of G that do not preserve H. In fact, we can achieve any permutation of the cyclic subgroups of order p using a 1-automorphism.