I'm essentially trying to figure out exactly what the title asks for. I've been scouring old Seminaires Henri Cartan and books by Stong to try to see exactly how to do this, but the combination of French and older terminology has made it rough going.

In particular, I've been able to work out (I think!) that, modulo 2, the map $$H_\ast(BSp;\mathbb{Z}/2)\to H_\ast(BU;\mathbb{Z}/2) $$ takes $y_{4k}$ to $(x_{2k})^2$, and that the map

$$H_\ast(U;\mathbb{Z}/2)\to H_\ast(SO;\mathbb{Z}/2)$$

takes $a_{2k-1}$ to certain polynomials $t_{2k-1}$ (the primitives) defined recursively as $$t_1=c_1,t_3=c_3+c_1c_2,\ldots,t_{2k-1}=\sum_{\substack{i+j=2k+1\\0\leq i<j}}c_ic_j$$ where the $a_{2k-1}$ and $c_k$ generate the homology groups. I'd really appreciate anything, from a complete exposition, to a reference, to a hint about something that I've missed or gotten wrong. I'm obviously especially interested in working out how to lift that last map to the classifying spaces (which would require completely knowing what the so-called "suspension" maps look like).

Thanks so much!

-Jon