Conjugate Dynamical Systems on C*-algebras

Let $(A, \alpha)$ and $(B, \beta)$ be C*-dynamical systems where $\alpha$ and $\beta$ are arbitrary *-endomorphisms. When $\alpha$ is injective or surjective, we show that the semicrossed products $A \times_\alpha \mathbb{Z}$ and $B \times_\beta \mathbb{Z}$ are isometrically isomorphic if and only if $(A, \alpha)$ and $(B, \beta)$ are outer conjugate. This conclusion also holds in various other cases as well.