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Let 
$\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) 
$\mathfrak{C}$ contains the trivial group, (ii) 
$\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by 
$\mathbb{Z}$. Note that 
$\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-
$\mathbb{Z}$ groups. Also, 
$\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this article, we show that if 
$\Gamma\in \mathfrak{C}$, then all strongly outer actions of Γ on the Razak–Jacelon algebra 
$\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szabó’s result for strongly self-absorbing C
$^*$-algebras.
