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Machine learning models now automate decisions in applications where we may wish to provide recourse to adversely affected individuals. In practice, existing methods to provide recourse return actions that fail to account for latent characteristics that are not captured in the model (e.g., age, sex, marital status). In this paper, we study how the cost and feasibility of recourse can change across these latent groups. We introduce a notion of group-level plausibility to identify groups of individuals with a shared set of latent characteristics. We develop a general-purpose clustering procedure to identify groups from samples. Further, we propose a constrained optimization approach to learn models that equalize the cost of recourse over latent groups. We evaluate our approach through an empirical study on simulated and real-world datasets, showing that it can produce models that have better performance in terms of overall costs and feasibility at a group level.