Approximately inner flows

When $\alpha$ is an approximately inner flow on a C${}^{\ast }$-algebra $A$ and commutes with an automorphism $\gamma$ of $A$ we may extend $\alpha$ to a flow $\bar{\alpha }$ on the crossed product $A{\times }_{\gamma }Z$ by setting ${\bar{\alpha }}_t(U)=U$ where $U$ is the canonical unitary implementing $\gamma$ in $A{\times }_{\gamma }Z$ and ask whether $\bar{\alpha }$ is also approximately inner or not. We will consider very specific examples of this type; some of which we can answer affirmatively.