I found a very strange isomorphism that I did not know have to prove.
Let $G$ be a finite discrete group. Let $Ad\rho_g(T)=\rho_g T\rho_g^*$ for all $T\in K(L^2(G))$ with $g\in G$ then: \begin{align*} (A\rtimes_\alpha G)\cong(A\otimes K(L^2(G)))^{\alpha\otimes Ad\rho} \end{align*} For which $A\otimes K(L^2(G))^{\alpha\otimes Ad\rho}$ is the fixed point algebra: \begin{align*} A\otimes K(L^2(G))^{\alpha\otimes Ad\rho}=\{a\otimes h\ |\ \alpha(a)\otimes Ad\rho_g(h)=a\otimes h\text{ for all }g\in G\} \end{align*} I do not know how to show this. Can anyone help? I found it [Yang] page 18.
[Yang]: Yang, Mingze. "Crossed products by finite groups acting on low dimensional complexes and applications." PhD diss., University of Saskatchewan, 1997.