We consider a crossed product of a unital simple separable nuclear stably finite Ƶ-stable C∗-algebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γ is elementary amenable, we show that the twisted crossed product C∗-algebra is Ƶ-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Ƶ are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C∗-algebras.
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