Quasiautomorphism-invariant not implies 1-automorphism-invariant

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., quasiautomorphism-invariant subgroup) need not satisfy the second subgroup property (i.e., 1-automorphism-invariant subgroup)
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Statement

It is possible to have a group $G$ and a subgroup $H$ of $G$ such that $H$ is a quasiautomorphism-invariant subgroup of $G$ (i.e., $H$ is invariant under all Quasiautomorphism (?)s of $G$) but is not a 1-automorphism-invariant subgroup of $G$ (i.e., $H$ is not invariant under all 1-automorphisms of $G$).

Related facts

• Characteristic not implies quasiautomorphism-invariant
• Center not is 1-automorphism-invariant

Facts used

1. Center is quasiautomorphism-invariant

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 $G$.

By fact (1), $H$ is quasiautomorphism-invariant in $G$. However, 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.