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Approximately inner flows

When $α$ is an approximately inner flow on a C$^*$-algebra $A$ and commutes with an automorphism $γ$ of $A$ we may extend $α$ to a flow $\barα$ on the crossed product $A\times_γZ$ by setting $\barα_t(U)=U$ where $U$ is the canonical unitary implementing $γ$ in $A\times_γZ$ and ask whether $\barα$ is also approximately inner or not. We will consider very specific examples of this type; some of which we can answer affirmatively.