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?

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?

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