Let me state first off that in this question I am most interested in lattice gauge theories, and not necessarily with Fermion couplings. But if Fermions and continuum gauge theories can also be addressed in the same breath, I am interested in them too.
I have seen many different gauge groups for both lattice gauge theories and continuum field theories---SU(N), O(N), and Z_N being the most common---but I do not know whether there is a systematic way to generate a gauge theory for an arbitrary group, or even whether it is sensible to talk about gauge theories for arbitrary groups. So my first question is: Can any group be a gauge group for some theory? If not, why? If so, to what extent is the structure of the gauge theory determined by the group?
Related to the last point, is there a "canonical" choice of action (or Boltzmann weight) associated to a given gauge group? For example, in this paper (eqns. 5 and 7) I see that they define a Boltzmann weight for an S_N lattice gauge theory in terms of a sum of characters of the group. Is such a prescription general?