We consider inclusions $\mathcal{ S}\leq \mathcal{ R}$ of discrete, probability measure-preserving orbit equivalence relations. In previous work with Ab\'{ e}rt-Fra\c{ c}zyk, we established the pointwise almost sure existence of the cospectral radius of a random walk on the $\mathcal{ R}$-classes. In this paper, we investigate the connections of this cospectral radius to the coamenability of the inclusion $\mathcal{ S}\leq \mathcal{ R}$. We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion.
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