Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and $A\rtimes_\alpha G$. Question 1: Is the canonical map $A\rtimes_\alpha \Gamma \to A\rtimes_\alpha G$ injective? Question 2: What about reduced crossed products $A\rtimes_{\alpha,r} \Gamma \to A\rtimes_{\alpha,r} G$?

For the universal case, I guess it's wrong even though I can not find a counterexample. But if we let $\alpha$ be the trivial action of $G$, then we only need to look at $A\otimes_{max} C^* (\Gamma) \subseteq A\otimes_{max} C^* (G) $.

For the reduced case, I guess it's should be the case. Just follows from the facts that the left regular representaion of $G$, restricted to $\Gamma$, is a multiple of the left regular representation of $\Gamma$ and $A\rtimes_{\alpha,r} \Gamma = C^*(\pi(A), 1\otimes \lambda(G))$, where $\pi: A\subseteq B(H) \rightarrow B(H\otimes l^2(G))$ and $\lambda$ is the left regular representaion of $G$. Do I make any mistake?