First, let me recall what an abstract HNN extension is. Let $G$ be an abstract group, $A, B < G$ be subgroups of $G$ and $\phi : A \to B$ be an isomorphisms. Then there is a group $H$ and an element $h \in H$ such that $G < H$ and for any $a \in A$ one has $\phi(a) = hah^{-1}$. Such an extension is, certainly, not unique, but there is a canonical way to construct one. The group $H$ obtained in this canonical way is called HNN extension of $G$ by $\phi$. More information can be found in a nice wikipedia article.

*My question is: for what classes of topological groups HNN extension is possible?*

More formally, let $G$ be a topological group, $A, B < G$ and $\phi : A \to B$ is an isomorphism in the category of topological groups. Is there a topological group $H$ that contains $G$ and such that $A$ and $B$ are conjugated inside $H$? If yes, what properties $H$ inherits from $G$, e.g., if $G$ is metrizable can $H$ be chosen metrizable?

Let me also mention that classical HNN construction uses free products and amalgamation of abstract groups. It is known that amalgamation of two Hausdorff topological groups over a closed subgroup may not be Hausdorff (though amalgamation, that appears in HNN extension is quite special, I don't know how bad it is).

The simplest case when $A$ is generated by one element already seems interesting.