  In this paper we continue our investigation of the global categorical
symmetries that arise when gauging finite higher groups and their higher
subgroups with discrete torsion. The motivation is to provide a common
perspective on the construction of non-invertible global symmetries in higher
dimensions and a precise description of the associated symmetry categories. We
propose that the symmetry categories obtained by gauging higher subgroups may
be defined as higher group-theoretical fusion categories, which are built from
the projective higher representations of higher groups. As concrete
applications we provide a unified description of the symmetry categories of
gauge theories in three and four dimensions based on the Lie algebra
$\mathfrak{so}(N)$, and a fully categorical description of non-invertible
symmetries obtained by gauging a 1-form symmetry with a mixed 't Hooft anomaly.
We also discuss the effect of discrete torsion on symmetry categories, based a
series of obstructions determined by spectral sequence arguments.
