  By the Fourier transformations, any group-invariant functions over finite
Abelian groups are transformed into group-invariant functions over the
character groups. In this paper, we calculate matrix elements of this
transformations under specific bases. More specifically, we deal with some
vector spaces over a finite field and linear actions. Then the matrix elements
under adequate bases are expressed by Krawtchouk or Affine q-Krawtchouk
polynomials. For calculations, we construct a commutative diagram which
combines two settings of group-invariant Fourier transformations. We apply it
to different sized transformations of each example, and solve it inductively.
We remark the matrix elements are related to the zonal spherical functions of
finite Gelfand pairs.
