Is there a classification of all equivariant structures of the Möbius line bundle $\ell\to S^1$?.

For example the antipodal action of $\mathbb{Z}/2\mathbb{Z}$ on $S^1$ cannot be lifted to the total space $\ell$ to get an equivariant structure. But what about the general case?

In particular, let $\phi_{\theta}$ be the irrational rotation of the circle by $\theta$. Can the action of $\mathbb{Z}$ on $S^1$ given by $n.x=\phi_{\theta}^n(x)$ be lifted to an action on the total space of the Möbius bundle to give us an equivariant bundle?