We introduce a method to make inference on the subgroups' sizes of a heterogeneous population using survey data, even in the presence of a single list. To this aim, we use Fisher's noncentral hypergeometric distribution, which allows us to account for the possibility that capture heterogeneity is related to key survey variables. We propose a Bayesian approach for estimating the population sizes posterior distributions, exploiting both extra-experimental information, e.g., coming from administrative data, and the computational efficiency of MCMC and ABC methods. The motivating case study deals with the size estimation of the population of Italian youngsters who are not employed one year after graduating by gender and degree program. We account for the possibility that surveys' response rates differ according to individuals' employment status, implying a not-at-random missing data scenario. We find that employed persons are generally more inclined to answer the questionnaire; this behavior might imply the overestimation of the employment rate.

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