Profinite rigidity for free-by-cyclic groups with centre

A free-by-cyclic group $F_N\rtimes_\phi\mathbb{Z}$ has non-trivial centre if and only if $[\phi]$ has finite order in ${\rm{Out}}(F_N)$. We establish a profinite ridigity result for such groups: if $\Gamma_1$ is a free-by-cyclic group with non-trivial centre and $\Gamma_2$ is a finitely generated free-by-cyclic group with the same finite quotients as $\Gamma_1$, then $\Gamma_2$ is isomorphic to $\Gamma_1$. One-relator groups with centre are similarly rigid. We prove that finitely generated free-by-(finite cyclic) groups are profinitely rigid in the same sense; the proof revolves around a finite poset $\mathbf{fsc}(G)$ that carries information about the centralisers of finite subgroups of $G$ -- it is a complete invariant for these groups. These results provide contrasts with the lack of profinite rigidity among surface-by-cyclic groups and (free abelian)-by-cyclic groups, as well as general virtually-free groups.