Approximately inner flows

When $\alpha$ is an approximately inner flow on a C$^*$-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.