On gauging finite subgroups

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\mathrm{\Gamma }$-symmetric theory. Depending on how anomalous $\mathrm{\Gamma }$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=\mathrm{\Gamma }\mathrm{/}A$ and a higher-form symmetry $\widehat{A}$ with a mixed anomaly, where $\widehat{A}$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\widehat{A}$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\widehat{A})$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.