[edit]
I'll try to be more precise.


In paper N.Nakamura, "Bauer–Furuta invariants under $Z\_2$-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 a $Z\_2$ action on $M$ lifts?


2) What about other groups (different than $Z\_2$)?


I'll be greatful also for general references on this topic.



Group action on spin^c 4-manifold.
This question is discussed to some extent in the following papers.


Bauer-Furuta invariants and Galois symmetries  

<http://dx.doi.org/10.1093/qmath/har021>


Characteristic cohomotopy classes for families of 4-manifolds  

<http://dx.doi.org/10.1515/forum.2010.027>


