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?