I know zero about the characteristic classes of finite group representations and would appreciate a reference. specifically, if I have a faithful representations of a finite group,G, in O(n) what can I say about the induced map on cohomology, P**(BO(n))-> H*(BG) ? I am mostly interested in Z or Z/2 coefficients so this would be Pontryagin classes and Stiefel-Whitney classes. For instance what are the induced maps of these two representations of Z/2? -1 reflection of the real line -1 0 0 -1 180 degree rotation of R^2. Can you show me an unoriented representation of the dihedral group of order 8 so that the induced map on Z/2 cohomology is not surjective?