Endomorph-dominating not implies automorph-conjugate

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., endomorph-dominating subgroup) need not satisfy the second subgroup property (i.e., automorph-conjugate subgroup)
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Statement

It is possible to have a group $G$ and a subgroup $H$ such that both the following hold:

• $H$ is an endomorph-dominating subgroup of $G$: For every endomorphism $\sigma$ of $G$, there exists $g\in G$ such that $\sigma (H)\leq gHg^{-1}$.
• $H$ is not an automorph-conjugate subgroup of $G$: There exists an automorphism $\sigma$ of $G$ such that $H$ and $\sigma (H)$ are not conjugate subgroups.

Proof

We take $G=\mathbb {Q} \rtimes (\mathbb {Q} ^{*})^{2}$ and $H$ as a copy of $\mathbb {Z}$ inside the base $\mathbb {Q}$.