By a mapping tori of $G$, I mean a semidirect product $G\rtimes\mathbb{Z}$, and by a trivial mapping tori I mean one isomorphic to $G\times\mathbb{Z}$.
If $G$ is finitely generated but not finitely presentable, clearly $G\times\mathbb{Z}$ is not finitely presentable. My question is: Does there exist a group $G$, with some property so as to be interesting, and where every other mapping torus is finitely presentable?
More precisely: Does there exist a finitely generated, non-finitely presentable group $G$ such that:
- $\operatorname{Out}(G)$ is not virtually cyclic, and
- for all $\phi\in\operatorname{Aut}(G)\setminus\operatorname{Inn}(G)$, the mapping torus $G\rtimes_{\phi}\mathbb{Z}$ is finitely presentable?
Condition (1) ensures that any answer isn't just a case of checking finitely many outer automorphisms (and there are examples with $\operatorname{Out}(G)$ finite and with $\operatorname{Out}(G)\cong\mathbb{Z}$, via Bumagin and Wise's version of Rips' construction). Note that $\operatorname{Out}(G)$ is considered rather than $\operatorname{Aut}(G)$, as two automorphisms of $G$ from the same outer automorphism class define isomorphic mapping tori.
My motivation here is that I was idly wondering if a group/class of groups I was looking at has this property. However, I do not know if any groups have this property, so this question is a sanity check (and seems interesting in it's own right). Moreover, I am interested to see what techniques could be used here - so multiple answers using a variety of techniques would be amazing!
