I'll try to be a compact Lie group acting on more precise.
In paper N.Nakamura, "Bauer–Furuta invariants under $M^4$ - riemannian four dimensional manifold with choosen Z_2$-actions" there is an assumption that $Z_2$ action "lifts to spin^c structure". What i think it means: $Spin^c$ structure is a principal $Spin^c$ bundle $\pi: P \to M$. A lift is (following Gottlieb) an action on $P$ such that $\pi$ is equivariant.
1) What are the conditions under which this a $Z_2$ action on $M$ liftsto spin^c?
Can anyone give some
2) What about other groups (different than $Z_2$)?
I'll be greatful also for general references ?on this topic.