  Let $K\leq H$ be two finite groups and let $C\leq A$ be two finite abelian
groups, with $H$ acting on $A$ as a group of isomorphisms admitting $C$ as a
$K$-invariant subgroup. We study the homogeneous space
$X\coloneqq\left(H\ltimes A\right)/\left(K\ltimes C\right)$ and determine the
decomposition of the permutation representation of $H\ltimes A$ acting on $X$.
We then characterize when this is multiplicity-free, that is, when
$\left(H\ltimes A,K\ltimes C\right)$ is a Gelfand pair. If this is the case, we
explicitly calculate the corresponding spherical functions. From our general
construction and related analysis, we recover Dunkl's results on the $q$-analog
of the nonbinary Johnson scheme.
