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?



Given a C-star dynamical system and a subgroup of the acting group, is the corresponding map on crossed product algebras necessarily an injection
The universal case is also injective. What one needs to show is that any covariant representation of $(A,\Gamma)$ extends to a covariant representation of $(A,G)$. This can be shown by using induced representation.


