Characteristic classes for spherical fibrations with fibre-preserving free group actions

Let G(H) be the monoid of H equivariant self maps of S(nV), the unit sphere of n copies of a finite dimension orthogonal representation V of a finite group H, stabilized over n in an appropriate way. Let SG(H) be the submonid of G(H) consisting of all degree 1 maps. If H₁ is a subgroup of H there is a natural forgetful map SG(H) → SG(H₁) and if Z is the center of H there is a natural action map BZ × SG(H) → SG(H) induced by the natural action of Z on H. The main results of this paper are the calculations of the Hopf algebra structures of H_*(SG(Z/pⁿ), Z/p) and H_*(BSG(Z/pⁿ), Z/p) for all n and all primes p, the calculations in homology of forgetful maps induced by the natural inclusions Z/p^n-1 → Z/pⁿ and, for H = Z/2, the calculation of the action map H_*(R P^∞, Z/2) ⊗ H_*(BSG(Z/2), Z/2) → H_* (BSG(Z/2), Z/2).