Let $\Gamma$ be a discrete group. If $0 \rightarrow I \rightarrow A \rightarrow \frac{A}{I} \rightarrow 0$ is a short exact $\Gamma$-algebras, then the sequence $0 \rightarrow I\rtimes \Gamma \rightarrow A \rtimes \Gamma \rightarrow \frac{A}{I}\rtimes \Gamma \rightarrow 0$ is also exact for every action $\alpha: \Gamma \rightarrow Aut(A)$ such that $\alpha_g(I))=I$ for all $g\in \Gamma$.
How can I show exactness in the middle?
