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 $G_1 \times \mathbb{Z} \cong G_2 \times \mathbb{Z}$.