Endomorph-dominating not implies automorph-conjugate

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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:

Proof

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