  We analyse the Higgs branch of 4d $\mathcal{N}=2$ SQCD gauge theories with
non-connected gauge groups $\widetilde{\mathrm{SU}}(N) = \mathrm{SU}(N)
\rtimes_{I,II} \mathbb{Z}_2$ whose study was initiated in arXiv:1804.01108. We
derive the Hasse diagrams corresponding to the Higgs mechanism using adapted
characters for representations of non-connected groups. We propose 3d
$\mathcal{N}=4$ magnetic quivers for the Higgs branches in the type $I$
discrete gauging case, in the form of recently introduced wreathed quivers, and
provide extensive checks by means of Coulomb branch Hilbert series
computations.
