  The enhancement of the spectrum of primordial comoving curvature perturbation
$\R$ can induce the production of primordial black holes (PBH) which could
account for part of present day dark matter. As an example of the effects of
the modification of gravity on the production of PBHs, we investigate the
effects on the spectrum of $\R$ produced by the modification of gravity in the
case of G-inflation, deriving the relation between the unitary gauge curvature
perturbation $\zeta$ and the comoving curvature perturbation $\R$, and
identifying a background dependent enhancement function $\E$ which can induce
large differences between the two gauge invariant variables. We use this
relation to derive an equation for $\R$, showing for the presence of a momentum
dependent effective sound speed (MESS), associated to the intrinsic entropy
which can arise in modified gravity theories, in agreement with the model
independent MESS approach to cosmological perturbations.
  When $\zeta$ is not constant in time it is different from $\R$, for example
on sub-horizon scales, or in models exhibiting an anomalous super-horizon
growth of $\zeta$, but since this growth cannot last indefinitely, eventually
they will coincide. We derive the general condition for super-horizon growth of
$\zeta$, showing that slow-roll violation is not necessary. Since the abundance
of PBHs depends on the statistics of the peaks of the comoving density
contrast, which is related to the spectrum of $\R$, it is important to take
into account these effects on the PBHs abundance in modified gravity theories.
