Outer automorphisms of HNN extensions of abelian groups

Let $G$ be a commensurating HNN extension of a finitely generated free abelian group. We characterise when $\mathrm{O}\mathrm{u}\mathrm{t}(G)$ is finitely generated and derive structural information about the group. We give soluble and non-soluble examples of commensurating HNN extensions of finitely generated free abelian groups with outer automorphism group not finitely generated.