  We consider an inverse seesaw mechanism of neutrino mass generation in which
the Standard Model is extended by $3+3$ (heavy) sterile states, and endowed
with a flavour symmetry $G_f$, $G_f=\Delta (3 \, n^2)$ or $G_f=\Delta (6 \,
n^2)$, and a CP symmetry. These symmetries are broken in a peculiar way, so
that in the charged lepton sector a residual symmetry $G_\ell$ is preserved,
while the neutral fermion sector remains invariant under the residual symmetry
$G_\nu=Z_2 \times CP$. We study the concrete setup, where the Majorana mass
term for three of the sterile states conserves $G_\nu$, while the remaining
mass terms (i.e. couplings of left-handed leptons and heavy sterile states, as
well as the Dirac-type couplings among the latter) do not break the flavour or
CP symmetry. We perform a comprehensive analysis of lepton mixing for different
classes of residual symmetries, giving examples for each of these, and study in
detail the impact of the additional sterile states on the predictions for
lepton mixing. We further confront our results with those obtained in the
model-independent scenario, in which the light neutrino mass matrix leaves the
residual symmetry $G_\nu$ intact. We consider the phenomenological impact of
the inverse seesaw mechanism endowed with flavour and CP symmetries, in
particular concerning effects of non-unitarity of the lepton mixing matrix
(which strongly constrain the parameter space of the scenario), prospects for
neutrinoless double beta decay and for charged lepton flavour violating
processes.
