Thanks for clarifying your question.

I rarely think of spin^c structures in terms of principal bundles. A map of a manifold lifts to a map of its principal $SO_n$ bundle if and only if the map is an orientation-preserving isometry. To go the additional step to lift to the principal $spin^c$-bundle you need to preserve the spin^c structure. So depending on how you want to think of spin^c structures there's various ways of thinking about this.

One is that a spin^c structure gives you an additional complex line bundle + further data. So your action has to act as a symmetry of this additional line bundle. Checking this is an entirely cohomological computation. On top of that, a spin^c structure means you have a spin structure on the direct sum of your tangent bundle and this complex line bundle. Again, checking your group action preserves this spin structure is cohomological in nature.

So if your group action is an involution like in the title of the paper you cite, the existence of the lift boils down to two rather simple cohomological computations. If you think of the spin^c structures on a manifold $M$ as being an affine space, your group acting on the manifold also acts on the set of all spin^c structures on the manifold, and that your particular spin^c structure has to be a fixed-point of this action.

If you have a particular example you're interested in it might make sense to just compute in that case.