| Summary: | 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 G?CP?ZS2×H?CP in the neutrino sector and GlCP?ZT3?HlCP in the charged lepton sector, leading to the prediction of no CP violation, namely ? CP and the Majorana phases ? 21 and ? 31 are all equal to either zero or ?. 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 ? CP ? ±?/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 ? CP ?=?0, ? or maximally broken with ? CP ?=?±?/2 and sharp predictions for the Majorana phases and neutrinoless double beta decay
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