  We discuss several bosonic topological phases in (3+1) dimensions enriched by
a global $\mathbb{Z}_2$ symmetry, and gauging the $\mathbb{Z}_2$ symmetry. More
specifically, following the spirit of the bulk-boundary correspondence,
expected to hold in topological phases of matter in general, we consider
boundary (surface) field theories and their orbifold. From the surface
partition functions, we extract the modular $\mathcal{S}$ and $\mathcal{T}$
matrices and compare them with $(2+1)$d toplogical phase after dimensional
reduction. As a specific example, we discuss topologically ordered phases in
$(3+1)$ dimensions described by the BF topological quantum field theories, with
abelian exchange statistics between point-like and loop-like quasiparticles.
Once the $\mathbb{Z}_2$ charge conjugation symmetry is gauged, the
$\mathbb{Z}_2$ flux becomes non-abelian excitation. The gauged topological
phases we are considering here belong to the quantum double model with
non-abelian group in $(3+1)$ dimensions.
