  We define gauge theories whose gauge group includes charge conjugation as
well as standard $\mathrm{SU}(N)$ transformations. When combined, these
transformations form a novel type of group with a semidirect product structure.
For $N$ even, we show that there are exactly two possible such groups which we
dub $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$. We construct the
transformation rules for the fundamental and adjoint representations, allowing
us to explicitly build four-dimensional $\mathcal{N}=2$ supersymmetric gauge
theories based on $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$ and understand
from first principles their global symmetry. We compute the Haar measure on the
groups, which allows us to quantitatively study the operator content in
protected sectors by means of the superconformal index. In particular, we find
that both types of $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$ groups lead to
non-freely generated Coulomb branches.
