On gauging Abelian extensions of finite and U(1) groups

We consider Abelian extensions of global symmetries of the form $A \to G \to K$, with $A$ finite (and similar higher-group structures). For a quantum field theory $\mathcal{T}$ with symmetry $G$, we compare gauging $G$ directly with gauging first $A$ and then $K$, and show that for finite Abelian groups and for $K \simeq U(1)$ the two procedures are equivalent as expected, $\mathcal{T}/G \simeq \mathcal{T}/A/K$. In the continuous case $K=U(1)$, after gauging the full extension, the dual symmetry $\widehat{\mathbb{Z}}_q^{(d-2)}$ fits into an extension characterizing the topological data of the magnetic $U(1)_m^{(d-3)}$ symmetry. This is better described using differential cohomology. We also briefly comment on the relation to symmetry fractionalization.