  We analyze orbifolds with discrete torsion of the ABJM theory by a finite
subgroup $\Gamma$ of $SU(2)\times SU(2)$ . Discrete torsion is implemented by
twisting the crossed product algebra resulting after orbifolding. It is shown
that, in general, the order $m$ of the cocycle we chose to twist the algebra by
enters in a non trivial way in the moduli space. To be precise, the M-theory
fiber is multiplied by a factor of $m$ in addition to the other effects that
were found before in the literature. Therefore we got a
$\mathbb{Z}_{\frac{k|\Gamma|}{m}}$ action on the fiber. We present a general
analysis on how this quotient arises along with a detailed analysis of the
cases where $\Gamma$ is abelian.
