  We discuss the origin of CP violation in settings with a discrete (flavor)
symmetry $G$. We show that physical CP transformations always have to be
class-inverting automorphisms of $G$. This allows us to categorize finite
groups into three types: (i) Groups that do not exhibit such an automorphism
and, therefore, in generic settings, explicitly violate CP. In settings based
on such groups, CP violation can have pure group-theoretic origin and can be
related to the complexity of some Clebsch-Gordan coefficients. (ii) Groups for
which one can find a CP basis in which all the Clebsch-Gordan coefficients are
real. For such groups, imposing CP invariance restricts the phases of coupling
coefficients. (iii) Groups that do not admit real Clebsch-Gordan coefficients
but possess a class-inverting automorphism that can be used to define a proper
(generalized) CP transformation. For such groups, imposing CP invariance can
lead to an additional symmetry that forbids certain couplings. We make use of
the so-called twisted Frobenius-Schur indicator to distinguish between the
three types of discrete groups. With $\Delta(27)$, $T^{\prime}$, and
$\Sigma(72)$ we present one explicit example for each type of group, thereby
illustrating the CP properties of models based on them. We also show that
certain operations that have been dubbed generalized CP transformations in the
recent literature do not lead to physical CP conservation.
