Endomorphisms of real Grassmannians that commute with Steenrod squares

For k E {2, 3} we classify endomorphisms of H*(Gk,n; Z2) that commute with Steenrod squares (here, Gk,n denotes the Grassmann manifold of k-dimensional subspaces in Rn+k). Additionally, for all positive integers k and n we classify endomorphisms of H*(Gk,n; Z2) that commute with Steen -rod squares and such that the image of each class is a polynomial in the (nonzero) class of H1(Gk,n; Z2).