For cyclic groups, you can in fact go further and calculate all of the Stiefel-Whitney classes similarly to how you calculate them for real projective spaces. You can find the latter in standard texts (eg Milnor-Stasheff, Characteristic Classes). If G is a cyclic group of order $m$, acting by a standard linear action, the quotient is a high-dimensional lens space. As such, there is an $m$-fold covering map from a sphere to the lens space, which embeds in the unit sphere bundle of a complex line bundle $\lambda$. The basic idea is to show that the tangent bundle, stabilized by adding trivial line bundles, splits as a sum of powers of $\lambda$; this leads directly to a formula for all Stiefel-Whitney (and Pontrjagin) classes.

I would imagine that a similar argument works for more general G, but some representation theory probably intervenes. In general, it's probably easier to try to solve the lifting problem algebraically as in José's answer above.

References for the cyclic case: J. Folkman, Equivariant maps of spheres into the classical groups, Mem. Amer . Math. Soc. 95 (1971).

Kwak, Jin Ho Some cyclic group actions on a homotopy sphere and the parallelizability of its orbit spaces. Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 807–818.

Ewing, J., S. Moolgavkar, L. Smith and R. S. Stong, Stable parallelizability of lens
spaces, J. of Pure and Applied Algebra, 10 (1977), 177-191.