We perform a comprehensive study of family symmetry models based on A4 combined with the generalised CP symmetry H CP. We investigate the lepton mixing parameters which can be obtained from the original symmetry A4 ⋊ H CP breaking to different remnant symmetries in the neutrino and charged lepton sectors. We find that only one case is phenomenologically viable, namely in the neutrino sector and in the charged lepton sector, leading to the prediction of no CP violation, namely δ CP and the Majorana phases α 21 and α 31 are all equal to either zero or π. We then propose an effective supersymmetric model based on the symmetry A4 ⋊ H CP in which trimaximal lepton mixing is predicted together with either zero CP violation or δ CP ≃ ± π/2 with non-trivial Majorana phases. An ultraviolet completion of the effective model yields a neutrino mass matrix which depends on only three real parameters. As a result of this, all three CP phases and the absolute neutrino mass scale are determined, the atmospheric mixing angle is maximal, and the Dirac CP can either be preserved with δ CP = 0, π or maximally broken with δ CP = ± π/2 and sharp predictions for the Majorana phases and neutrinoless double beta decay.