  We define an equivariant and equicovariant versions of the notion of module
nuclearity. More precisely, for a discrete group $\Gamma$ and operator
$\mathcal A$-$\Gamma$-(co)module $\mathcal B$, $\mathcal E$ over a
$\Gamma$-C$^*$-algebra $\mathcal A$, we define $\mathcal E$-$\Gamma$-nuclearity
of $\mathcal B$, as an equivariant version of the notion of $\mathcal
E$-nuclearity, in which the identity map on $\mathcal B$ is required to be
approximately factored through matrix algebras on $\mathcal E$ with module
structures coming both from the original module structure of $\mathcal E$ and
the $\Gamma$-action on $\mathcal E$. For trivial actions of $\Gamma$, this is
shown to reduce to the notion of module nuclearity, introduced and studied by
the first author. As a concrete example, for a discrete group $\Gamma$ acting
amenably on a unital C$^*$-algebra $\mathcal A$, we show that the reduced
crossed product $\mathcal A\rtimes_{r} \Gamma$ is $\mathcal
A$-$\Gamma$-nuclear. Conversely, if $\mathcal A$ is a nuclear C$^*$-algebra
with a $\Gamma$-invariant state $\rho$ and $\mathcal A\rtimes_{r} \Gamma$ is
$\mathcal A$-$\Gamma$-nuclear, then we deduce that $\Gamma$ is amenable. We
show that when $\mathcal A\rtimes_{r} \Gamma$ is $\mathcal A$-$\Gamma$-nuclear
and $\mathcal A$ has the completely bounded approximation property (resp., is
exact), then so is $\mathcal A\rtimes_{r} \Gamma$. We prove similar results for
$\mathcal A\rtimes_{r} \Gamma$, regarded as an $\mathcal A$-$\Gamma$-comodule.
