Suppose $G$ is a Lie group acting properly on a (non-compact) manifold $M$ with compact orbit space. Then there is an induced action of $G$ on $C_0(M)$, the continuous $\mathbb{C}$-valued functions on $M$ vanishing at infinity, and we can form the crossed product algebra $C_0(M)\rtimes G$.
I was told that there is an isomorphism
$$C_0(M)\rtimes G\cong C(M,\mathcal{K}(L^2(G)))^G,$$
where $\mathcal{K}(L^2(G))$ denotes the $C^*$-algebra of compact operators on the Hilbert space $L^2(G)$, and the superscript $G$ means $G$-invariant functions. At first I didn't care to ask what the actual map would be, since so often it is the case that the "obvious map" turns out to be the right one; but in this case I can't seem to figure out what the obvious map should be.
Question: What should the isomorphism be? Is there a reference where I can read more about this isomorphism?