a basic question in crossed product for compact group action I am quite new into crosssed product of Fréchet algebras or C$^*$-algebras. So if the question is too basic please excuse me.
Suppose we have two Fréchet algebras or C$^*$-algebras $A$ and $B$ and they are isomorphic by a map $\phi$. Moreover $G$ be a compact Lie group acting on $A$ and $B$ by $\alpha$ and $\beta$ respectively. When can we say that $\phi$ extends to a isomorphism of the crossed products $A\rtimes_{\alpha} G$ and $B\rtimes_{\beta} G$?

A sufficient condition is that $\phi$ intertwines the actions. Also, note that saying that $\phi$ "extends" to an isomorphism on the crossed products is slightly incorrect, since in general there is no inclusion $A\to A\rtimes G$.
