  In this work, we study multiplicity-free induced representations of finite
groups. We analyze in great detail the structure of the Hecke algebra
corresponding to the commutant of an induced representation and then specialize
to the multiplicity-free case. We then develop a suitable theory of spherical
functions that, in the case of induction of the trivial representation of the
subgroup, reduces to the classical theory of spherical functions for finite
Gelfand pairs. \par We also examine in detail the case when we induce from a
normal subgroup, showing that the corresponding harmonic analysis can be
reduced to that on a suitable Abelian group. \par The second part of the work
is devoted to a comprehensive study of two examples constructed by means of the
general linear group GL$(2,\mathbb{F}_q)$, where $\mathbb{F}_q$ is the finite
field on $q$ elements. In the first example we induce an indecomposable
character of the Cartan subgroup. In the second example we induce to
GL$(2,\mathbb{F}_{\!q^2})$ a cuspidal representation of GL$(2,\mathbb{F}_q)$.
