<p>Hypergraphs are the most general model for complex networks involving group interactions. Taking the ideas of path homology from Grigor’yan et al. ((Homologies of path complexes and digraphs. <a href="http://arxiv.org/abs/1207.2834">arXiv:1207.2834</a>, 2013; Torsion of digraphs and path complexes. <a href="http://arxiv.org/abs/2012.07302v1">arXiv:2012.07302v1</a>, 2020; Pure Appl Math Q 10(4):619–674 2014; Asian J Math 15(5):887–932, 2015; J Math Sci 248(5):564–599, 2020), Bressan et al. (Asian J Math 23(3):479–500, 2019) introduced embedded homology of hypergraphs, which has leaded to successful applications in protein-ligand binding network (Liu et al. in Brief Bioinform. <a href="https://doi.org/10.1093/bib/bbaa411">https://doi.org/10.1093/bib/bbaa411</a>, 2021; Brief Bioinform 22(5):bbab127, 2021). A fundamental question arising from practical applications is about the stability of the persistent embedded homology of hypergraphs. In this paper, we prove the stability of the persistent embedded homology as well as the persistent homology of the associated simplicial complex with respect to perturbations of the filtration on a hypergraph. We apply the persistent homology methods to morphisms of hypergraphs and prove the stability with respect to perturbations of the filtrations. We prove the constancy of the persistent Betti numbers under some conditions on the simple-homotopy types of hypergraphs.</p>