Homogeneous spaces of semidirect products and finite Gelfand pairs

Let $K\le H$ be two finite groups and let $C\le A$ 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 $X(H⋉A)\mathrm{/}(K⋉C)$ and determine the decomposition of the permutation representation of $H⋉A$ acting on $X$. We then characterize when this is multiplicity-free, that is, when $(H⋉A,K⋉C)$ 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.