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We present a criterion for multiplicity-freeness of the decomposition of therestriction Res$^G_H(\rho_1 \otimes \rho_2)$ of the Kronecker product of twogeneric irreducible representations $\rho_1, \rho_2$ of a finite group $G$ withrespect to a subgroup $H \leq G$. This constitutes a generalization of a wellknown criterion due to Mackey (which corresponds to the case $H = G$). Thecorresponding harmonic analysis is illustated by detailed computations on theClifford groups $G={\mathbb{CL}}(n)$, together with the subgroups$H={\mathbb{CL}}(n-1)$, for $n \geq 1$, which lead to an explicit decompositionof the restriction of Kronecker products.

Mackey's criterion for subgroup restriction of Kronecker products and harmonic analysis on Clifford groups