The following question is particularly interesting for me:

Does the natural map $Gr(3,5)\to Gr(3,6)$ induce a surjection
$$H^4(Gr(3,6),\mathbb{Z})\to H^4(Gr(3,5),\mathbb{Z})?$$

Here $Gr(k,n)$ means the real grassmannian of rank $k$.