This is a thought and not a proper answer but since we do not know what dark matter is this question is a bit point-less if you may. What I mean is that it depends on your model. If you take DM to be a supersymmetric particle, like the neutralino, then it can be involved in processes of the weak interaction according to the various MSSM-like models (and extensions of it of course). If it is the axion I am not quite sure about it since, again, it depends in the axion models. For example here (http://www.physics.ntua.gr/cosmo07/UniNet/Talks%20Thursday/Armillis.pdf) a PQ-like axion is coupled to some Higgses. MND I think is all gravitational but someone else, much more expert than myself, would have to confirm this.
I hope this helps a little bit.
*I forgot to mention the case of sterile neutrinos which in the models I am aware of they are not charged under any of the SM gauge groups and thus would not be involved in any weak interactions directly but byproducts of interactions among those would. See Shapsonikov's (among others) work for more details.