  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 $n=2,4,6,8.$
