  Let $A$ be a simple separable nuclear monotracial C$^*$-algebra, and let
$\alpha$ be an outer action of a finite abelian group $\Gamma$ on $A$. In this
paper, we show that $\alpha\otimes \mathrm{id}_{\mathcal{W}}$ on
$A\otimes\mathcal{W}$ is approximately representable if and only if the
characteristic invariant of $\tilde{\alpha}$ is trivial, where $\mathcal{W}$ is
the Razak-Jacelon algebra and $\tilde{\alpha}$ is the induced action on the
injective II$_1$ factor $\pi_{\tau_{A}}(A)^{''}$. As an application of this
result, we classify such actions up to conjugacy and cocycle conjugacy. In
particular, we show the following: Let $A$ and $B$ be simple separable nuclear
monotracial C$^*$-algebras, and let $\alpha$ and $\beta$ be outer actions of a
finite abelian group $\Gamma$ on $A$ and $B$, respectively. Assume that the
characteristic invariants of $\tilde{\alpha}$ and $\tilde{\beta}$ are trivial.
Then $\alpha\otimes \mathrm{id}_{\mathcal{W}}$ and $\beta\otimes
\mathrm{id}_{\mathcal{W}}$ are conjugate (resp. cocycle conjugate) if and only
if $\tilde{\alpha}$ on $\pi_{\tau_{A}}(A)^{''}$ and $\tilde{\beta}$ on
$\pi_{\tau_{B}}(B)^{''}$ are conjugate (resp. cocycle conjugate). We also
construct the model actions.
