Let $G=\langle H, t; A^t=B\rangle$ by an $HNN$-extension of $H$, $A$ and $B$ isomorpic subgroups of $H$ where conjugation by $t$ induces the isomorphism.


Assuming $H$ is a finite group it is a well-known consequence of Britton's Lemma that if $\hat{H}$ is a subgroup of $G$ with $\hat{H}\cong H$ then $\hat{H}$ is conjugate to $H$ in $G$.


I was wondering under what other conditions on $H$ this result would hold. Specifically,



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> What conditions can we place on $H$ such that if $\hat{H}$ is a subgroup of $G$ with $\hat{H}\cong H$ then $\hat{H}$ is conjugate to a subgroup of $H$.
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When are isomorphic copies of the base group in an $HNN$-extension subgroups of the base group (up to conjugacy)?
If H is a finitely generated torsion group, or more generally has Serre's property FA (fixed point property on trees) then what you want is true. Indeed H will have a fixed point in the Bass-Serre tree of the HNN extension. Since all vertex stabilizers are conjugate on this case, you are done. 


