 Solution

Choose a vacuum order parameter ${\Phi }_0$ and let $H$ be its stabilizer in $G$. Acting with $G$ produces all symmetry-related minima; two elements produce the same minimum precisely when they differ by an element of $H$. The vacuum manifold is thereforeThis assumes a single transitive orbit of minima; additional independent vacuum sectors would require more than one orbit. For a local gauge symmetry, $G/H$ describes the vacuum directions used in finite-energy boundary conditions and defect classification, rather than distinct observable vacua after quotienting by every gauge redundancy.