related by $${{\operatorname{rc}}}(A) \leq {{\operatorname{rc}}}\big(C^*(G, A, \alpha)\big) \leq {{\operatorname{rc}}}(A^{\alpha}).$$ In fact, we prove a much stronger result, relating the Cuntz semigroups (see Theorem \[WC\_injectivity\] and Theorem \[WC\_plus\_injectivity\]). We show that the inclusion of $A$ in $C^*(G, A, \alpha)$ induces an isomorphism from ${{\operatorname{Cu}}}(A)$ to its range in ${{\operatorname{Cu}}}(C^*(G, A, \alpha))$ if $\alpha$ is a strictly approximate inner action of a finite group $G$ on a unital C\*-algebra $A$. It induces an isomorphism from ${{\operatorname{Cu}}}_+ (A) \cup \{0\}$ to its range if $\alpha$ is a tracially strictly approximately inner action of a finite group $G$ on an infinite-dimensional simple unital stably finite C\*-algebra $A$.

Suppose further that $G$ is abelian, $\alpha$ is tracially approximately representable, and $C^*(G, A, \alpha)$ is simple. Let $\widehat{\alpha}$ be the dual action of $\widehat{G}$ on $C^*(G, A, \alpha)$. Then $\widehat{\alpha}$ has the tracial Rokhlin property by Theorem 3.11 of [@Ph11]. Therefore our results are immediate from Takai duality [@Tak75] and the main results in [@AGP19]. But the hypothesis in our results is only that the action is tracially strictly approximately inner. Also, the group in our work need not to be abelian. Thus, our results generalize the tracial Rokhlin property case of [@AGP19]. Note that the 