  We introduce gauge theories based on a class of disconnected gauge groups,
called principal extensions. Although in this work we focus on 4d theories with
N=2 SUSY, such construction is independent of spacetime dimensions and
supersymmetry. These groups implement in a consistent way the discrete gauging
of charge conjugation, for arbitrary rank. Focusing on the principal extension
of SU(N), we explain how many of the exact methods for theories with 8
supercharges can be put into practice in that context. We then explore the
physical consequences of having a disconnected gauge group: we find that the
Coulomb branch is generically non-freely generated, and the global symmetry of
the Higgs branch is modified in a non-trivial way.
