  Given a finite group $G$ and a subgroup $K$, we study the commutant of
$\text{Ind}_K^G\theta$, where $\theta$ is an irreducible $K$-representation.
After a careful analysis of Frobenius reciprocity, we are able to introduce an
orthogonal basis in such commutant and an associated Fourier transform. Then we
translate our results in the corresponding Hecke algebra, an isomorphic algebra
in the group algebra of $G$. Again a complete Fourier analysis is developed
and, as particular cases, we obtain some results of Curtis and Fossum on the
irreducible characters of Hecke algebras. Finally, we develop a theory of
Gelfand-Tsetlin bases for Hecke algebras.
