  We continue the program of constructing (pre)modular tensor categories from
3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and
then mathematically studied by Cui-Qiu-Wang. An important structure involved is
a collection of certain $\text{SL}(2, \mathbb{C})$ characters on a given
manifold which serve as the simple object types in the corresponding category.
Chern-Simons invariants and adjoint Reidemeister torsions play a key role in
the construction, and they are related to topological twists and quantum
dimensions, respectively, of simple objects. The modular $S$-matrix is computed
from local operators and follows a trial-and-error procedure. It is currently
unknown how to produce data beyond the modular $S$- and $T$-matrices. There are
also a number of subtleties in the construction which remain to be solved. In
this paper, we consider an infinite family of 3-manifolds, that is, torus
bundles over the circle. We show that the modular data produced by such
manifolds are realized by the $\mathbb{Z}_2$-equivariantization of certain
pointed premodular categories. Here the equivariantization is performed for the
$\mathbb{Z}_2$-action sending a simple (invertible) object to its inverse, also
called the particle-hole symmetry. It is our hope that this extensive class of
examples will shed light on how to improve the program to recover the full data
of a premodular category.
