  Let $A$ and $B$ be simple separable nuclear monotracial C$^*$-algebras, and
let $\alpha$ and $\beta$ be strongly outer actions of a countable discrete
amenable group $\Gamma$ on $A$ and $B$, respectively. In this paper, we show
that $\alpha\otimes\mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$ and
$\beta\otimes\mathrm{id}_{\mathcal{W}}$ on $B\otimes\mathcal{W}$ are cocycle
conjugate where $\mathcal{W}$ is the Razak-Jacelon algebra. Also, we
characterize such actions by using the fixed point subalgebras of Kirchberg's
central sequence C$^*$-algebras.
