In this paper we consider a bootstrap class $mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $Gammainmathfrak C$ and any strongly self-absorbing C*-algebra $mathcal D$: (1) There is a unique strongly outer $Gamma$-action on $mathcal D$ up to (very strong) cocycle conjugacy. (2) If $alpha: Gammacurvearrowright A$ is a strongly outer action on a separable, unital, nuclear, simple, $mathcal D$-stable C*-algebra with at most one trace, then it absorbs every $Gamma$-action on $mathcal D$ up to (very strong) cocycle conjugacy. In fact we establish more general relati
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