Coherent configurations over copies of association schemes of prime order

Let $G$ be a group acting faithfully and transitively on $\Omega_i$ for $i=1,2$. A famous theorem by Burnside implies the following fact: If $|\Omega_1|=|\Omega_2|$ is a prime and the rank of one of the actions is greater than two, then the actions are equivalent, or equivalently $|(\alpha,\beta)^G|=|\Omega_1|=|\Omega_2|$ for some $(\alpha,\beta)\in \Omega_1\times \Omega_2$. In this paper we consider a combinatorial analogue to this fact through the theory of coherent configurations, and give some arithmetic sufficient conditions for a coherent configuration with two homogeneous components of prime order to be uniquely determined by one of the homogeneous components.