Quasiautomorphism-invariant not implies 1-automorphism-invariant

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 fact about::quasiautomorphisms 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) uses::Center is quasiautomorphism-invariant

Proof
Let $$p$$ be an odd prime, let $$G$$ be the particular example::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.