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Let $G={*}_{j=1}^{q+1}{G}_{n_j+1}$ be the product of $q+1$ finite groups each having order $n_j+1$ and let $\mu$ be the probability measure which takes the value $p_j/n_j$ on each element of $G_{n_j+1}\setminus \left\{e\right\}$. In this paper we shall describe the point spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers $n_j$. We also compute the continuous spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ in several cases. A family of irreducible representations of $G$, parametrized on the continuous spectrum of $\mu$,...