Transitive action on non orientable $ M $ lifts to orientable double cover

Question

Suppose that $ M $ is non orientable with transitive action by a Lie group $ G $. Does that imply that some Lie group $ G' $ acts transitively on the orientable double cover $M'$?

This is true for compact dimension 2: the Klein bottle is an $ SE_2 $ manifold and so is the torus. The projective plane is an $ SO_3 $ manifold and so is the sphere.

Morally I think the answer should be yes in general since an orientable double cover should be better behaved /at least as well behaved as the original non orientable manifold.

Answer 1

The answer is yes given here

https://mathoverflow.net/questions/410565/transitive-action-on-non-orientable-m-lifts-to-orientable-double-cover

Although you can always lift a transitive action to the orientable double cover, a transitive action on the orientable double cover does not always descend. Consider for example a nonorientable three manifold double covered by the three torus.