I have a (maybe silly) question:
A profinite HNN-extension $H(G)$ of a profinite group $G$ is defined as the profinite completion of an abstract HNN-extension of $G$. Here is my question.
If $G = \overline{\langle X \rangle}$, is $H(G)$ the profinite completion of an HNN-extension of $\langle X \rangle$?
This question is motivated by the following problem: in the abstract case, the natural map of a group $G$ to its HNN-extension $G^*$ is a monomorphism, but in the profinite case this may not true. So, when the map from $G$ to $H(G)$ is a monomorphism, we say that the profinite HNN-extension is proper. My objective is to find some example of a non-proper profinite HNN-extension. If $\widehat{\Bbb{Z}}$ is the profinite completion of $\Bbb{Z}$, then $\widehat{\Bbb{Z}} = \overline{\langle g \rangle}$ where $g$ is a generator of $\Bbb{Z}$. We know that $B(2,3)$ (Baumslag-Solitar group) can be viwed as an HNN-extension of $\Bbb{Z}$ and $B(2,3)$ is not residually finite (because it is finitely generated and non-Hopfian by Britton's lemma). So if $\widehat{B(2,3)}$ (the profinite completion of $B(2,3)$) is a profinite HNN-extension of $\widehat{\Bbb{Z}}$, it would be an example.