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,

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 $H$.

(The title should have "...up to conjugacy" at the end of it - but I felt it was getting a bit long!)

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,

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 $H$.