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. 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, $\mathcal{T}\mathrm{/}G\simeq \mathcal{T}\mathrm{/}A\mathrm{/}K$. 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.