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

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

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