Suppose $A$ and $B$ are two isomorphic $C^*$-algebras, i.e, there is a $^*$-isomorphism from $A$ to $B$ and $G$ is a group acting on both the $C^*$-algebras by automorphism. If $\alpha$ and $\beta$ denote the action of $G$ on $A$ and $B$ respectively, then are the corresponding crossed product $C^*$-algebras $A\rtimes_{\alpha} G$ and $B\rtimes_{\beta} G$ isomorphic? If, moreover, $G$ and $H$ are two isomorphic groups acting on $A$ and $B$ respectively by automorphism, then are the corresponding crossed product $C^*$-algebras $A\rtimes_{\alpha}G$ and $B\rtimes_{\beta}H$ isomorphic?
This is a repost from Mathoverflow: https://mathoverflow.net/questions/396774/isomorphism-of-crossed-product-c-algebras
No. Let $A=B=\mathbb C^2$, $G=\mathbb Z_2$. Take $\alpha$ to be the trivial action, $\alpha(g)x=x$; and $\beta(1)(a,b)=(b,a)$. Then $$ A\rtimes_\alpha G=\mathbb C^4,\qquad B\rtimes_\beta G=M_2(\mathbb C). $$