  We deal with the concrete spectral analysis of an invariant magnetic
Schr\"odinger operator acting on one dimensional $L^2$-mixed automorphic
functions with respect to given equivariant pair $ (\rho,\tau) $ and given
discrete subgroup of the semi-direct group $U(1)\ltimes\mathbb{C}$. This will
be carried out by means of a lifting theorem to the classical automorphic
functions associated with specific pseudo-character. We also provided a partial
characterization of the equivariant pairs relative to our setting and discuss
possible generalization to higher dimensions.
