Endomorphism rings of permutation modules over maximal Young subgroups

Let K be a field of characteristic two, and let λ be a two-part partition of some natural number r. Denote the permutation module corresponding to the (maximal) Young subgroup Σλ in Σr by Mλ. We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra SK(λ)=1λSK(2,r)1λ=EndKΣr(Mλ) of the Schur algebra SK(2,r). These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.