Natural map $C^*(G) \to M(A\rtimes G)$

The answer is no. For a counter example, take an amenable action $\alpha$ of a non-amenable discrete group G on a unital C*-algebra $A$. Then $A\rtimes_\alpha G$ coincides with the reduced crossed product and hence the standard conditional expectation is faithful. If your map were injective, the standard conditional expectation on $C^*(G)$ would also be faithful, but it isn't.