Perturbed flavour symmetries and predictions of CP violating phase $\delta$

It is known that the imposition of a class of residual $Z_2\times Z_2$ symmetries on the neutrino mass matrix $M_\nu$ and a residual symmetry $Z_n$ ($n\geq 3$) on the Hermitian combination $M_lM_l^\dagger$ of the charged lepton mass matrix leads to a universal prediction of vanishing Dirac CP phase $\delta$ if these symmetries are embedded in $\Delta(6 n^2)$ groups and if the leptonic doublets transform as a 3 dimensional irreducible representation of the group. The Majorana phases remain arbitrary but they can also be determined in $\Delta(6 n^2)$ by imposing generalized CP symmetry (GenCP) consistent with the $\Delta(6 n^2)$ group. We investigate the effects of adding general perturbations on these predictions assuming that perturbations break the $Z_2\times Z_2$ symmetry completely but preserve GenCP. It is found that if the residual symmetries predict the tri-bimaximal mixing (TBM) among leptons and specific CP conserving values for the Majorana phases then addition of the above perturbations always lead to a neutrino mass matrix invariant under the $\mu$-$\tau$ reflection symmetry in the flavour basis with the result that perturbations turn the vanishing $\delta$ into maximal value $\pm \frac{\pi}{2}.$ One gets non-vanishing but generally large $\delta$ if the predicted zeroeth order mixing deviates from TBM and/or the predicted Majorana phases are non-trivial. We systematically investigate effects of perturbations in such situations and work out the predicted $\delta$ for four of the lowest $\Delta(6 n^2)$ groups with $n=2,4,6,8.$