  We investigate deformations of $\mathbb{Z}_2$ orbifold singularities on the
toroidal orbifold $T^6/(\mathbb{Z}_2\times\mathbb{Z}_6)$ with discrete torsion
in the framework of Type IIA orientifold model building with intersecting
D6-branes wrapping special Lagrangian cycles. To this aim, we employ the
hypersurface formalism developed previously for the orbifold
$T^6/(\mathbb{Z}_2\times\mathbb{Z}_2)$ with discrete torsion and adapt it to
the $\mathbb{Z}_2\times\mathbb{Z}_6\times\Omega\mathcal{R}$ point group by
modding out the remaining $\mathbb{Z}_3$ subsymmetry and the orientifold
projection $\Omega\mathcal{R}$. We first study the local behaviour of the
$\mathbb{Z}_3\times\Omega\mathcal{R}$ invariant deformation orbits under
non-zero deformation and then develop methods to assess the deformation effects
on the fractional three-cycle volumes globally. We confirm that D6-branes
supporting USp(2N) or SO(2N) gauge groups do not constrain any deformation,
while deformation parameters associated to cycles wrapped by D6-branes with
U(N) gauge groups are constrained by D-term supersymmetry breaking. These
features are exposed in global prototype MSSM, Left-Right symmetric and
Pati-Salam models first constructed in arXiv:1509.00048 and arXiv:1409.1236,
for which we here count the number of stabilised moduli and study flat
directions changing the values of some gauge couplings. Finally, we confront
the behaviour of tree-level gauge couplings under non-vanishing deformations
along flat directions with the one-loop gauge threshold corrections at the
orbifold point and discuss phenomenological implications, in particular on
possible LARGE volume scenarios and the corresponding value of the string scale
$M_{\text{string}}$, for the same global D6-brane models.
