  We construct models of leptons based on $S_4$ family symmetry combined with a
generalised CP symmetry $H_{CP}$. We show how the flavon potential can
spontaneously break the symmetry $S_4 \rtimes H_{CP}$ down to $Z_2 \times
H^{\nu}_{CP}$ in the neutrino sector, where the choice of preserved CP symmetry
$H^{\nu}_{CP}$ is controlled by free (real) parameters in the flavon potential.
We propose two realistic models of this kind, one at the effective level and
one at the renormalisable level. Both models predict trimaximal lepton mixing
with CP being either fully preserved or maximally broken, with the intermediate
possibility forbidden by the structure of the models.
