Profinite rigidity properties of central extensions of 2-orbifold groups

We extend Wilkes’ results on the profinite rigidity of SFSs to the setting of central extensions of 2-orbifold groups with higher-rank centre. We prove that both rigid and non-rigid phenomena arise in this setting and that the non-rigid phenomena are transient in the sense that if ${\widehat{G}}_1\cong {\widehat{G}}_2$, then $G_1\times \mathbb{Z}\cong G_2\times \mathbb{Z}$.