It is well-known that, if I have a two-dimensional quantum field theory, $\mathcal{Q}$, with a finite, abelian, non-anomalous global symmetry $\Gamma$, and I gauge it, the resulting theory (which I denote $\mathcal{Q}/\!\!/\Gamma$) has the Pontryagin dual group, $\widehat{\Gamma}:=\mathrm{Hom}(\Gamma,\mathrm{U}(1))$, as a global symmetry. This is known as quantum symmetry or Ponryagin dual symmetry.
Relatively recently it was understood that the story is quite a bit more general. In $d$ dimensions with a $p$-form symmetry, the dual symmetry is a $(d-p-2)$-form symmetry, while if $\Gamma$ is finite but non-abelian, the Pontryagin dual symmetry is a non-invertible symmetry, usually denoted as $\mathrm{Rep}(\Gamma)$ or variations thereof depending on the dimensionality and/or other data that specify the symmetry category.
My question is, how common is it, actually, in a generic gauge theory?
For that, let's imagine the following scenario. I have a pure $G$-gauge theory where $G$ is any group (I could specialise to a subcategory of groups, such as connected or simply connected, if technically necessary, but I think it's fine as is). I want to think about it as starting off with the trivial theory with $G$-symmetry and gauging $G$, so I'll denote it $\bullet/\!\!/G$. In this type of gauge theories, there isn't necessarily Pontryagin dual symmetry for all of $G$ because the gauging involves summing over non-flat bundles (for the continuous pieces of $G$) and hence it is not an invertible process -- which Pontryagin duality (or more generally Tannaka-Krein duality and generalisations) would imply. However what I could do, starting with the trivial $G$-symmetric theory, $\bullet$, is choose any finite (probably need normal) subgroup $\Gamma\subset G$ and gauge $G/\Gamma$ to get $\bullet/\!\!/(G/\Gamma)$, which still has global symmetry $\Gamma$, since I didn't gauge it. Then gauge $\Gamma$, to get $\left[\bullet/\!\!/(G/\Gamma)\right]/\!\!/\Gamma \cong \bullet/\!\!/G$. Since in the last step I only gauged a finite group, I expect to get back a Pontryagin dual global symmetry, $\widehat{\Gamma}=\mathrm{Rep}(\Gamma)$.
The above argument implies the following:
Any pure $G$-gauge theory has a $(d-2)$-form Pontryagin dual global symmetry $\widehat{\Gamma}_\text{max}=\mathrm{Rep}(\Gamma_\text{max})$, where $\Gamma_\text{max}$ is the maximal (normal?) finite subgroup of $G$.
Is this true? It seems a bit too strong to be true. If not, what is wrong with the argument?
Moreover, in a simple example, this is not how it seems to work. Let's take $G=\mathrm{SU}(2)$ and $\Gamma= Z(\mathrm{SU}(2))\cong\mathbb{Z}_2$. Then $G/\Gamma\cong \mathrm{SO}(3)$ and $\bullet/\!\!/(G/\Gamma)$ is the pure $\mathrm{SO}(3)$-gauge theory. To go to the $\mathrm{SU}(2)$-gauge theory one gauges the centre $\mathbb{Z}_2$ as a one-form symmetry! Which would result in a $\widehat{\mathbb{Z}}_2$ dual symmetry that is a $(d-3)$-form symmetry, instead of a $(d-2)$-form symmetry as the above argument would suggest. Or could one gauge $\mathbb{Z}_2$ either as a zero-form symmetry (which acts on nothing in $\bullet/\!\!/\mathrm{SO}(3)$), or as a one-form symmetry and obtain different quantum symmetries in either case?