Discrete Gauging of 6d SCFTs and Wreathed 3d $\mathcal{N}=4$ Quivers

We study the Higgs branch moduli space of certain 6d $(1,0)$ SCFTs after gauging their ${\mathbb{Z}}_2$ Green-Schwarz automorphism. We explain how to read the flavor symmetry of such SCFTs directly from the 6d construction, and we confirm the expectation by computing the Coulomb branch Hilbert series of their ${\mathbb{Z}}_2$-wreathed 3d $\mathcal{N}=4$ magnetic quiver. To perform the latter computation, we explicitly introduce a methodology to determine such Hilbert series for ${\mathbb{Z}}_2$-wreathed orthosymplectic quivers.