Affinity of permutations of F2n

It was conjectured that if n   is even, then every permutation of F2n is affine on some 2-dimensional affine subspace of F2n. We prove that the conjecture is true for n=4n=4, for quadratic permutations of F2n and for permutation polynomials of F2nF2n with coefficients in F2n/2F2n/2. The conjecture is actually a claim about (AGL(n,2),AGL(n,2))(AGL(n,2),AGL(n,2))-double cosets in permutation group S(F2n) of F2n. We give a formula for the number of (AGL(n,2),AGL(n,2))(AGL(n,2),AGL(n,2))-double cosets in S(F2n) and classify the (AGL(4,2),AGL(4,2))(AGL(4,2),AGL(4,2))-double cosets in S(F24).