We show that if N and M are transitive models of ZFA such that N $\subseteq$ M, N and M have the same kernel and same set of atoms, and M $\models$ AC, then N is a Fraenkel-Mostowski-Specker (FMS) submodel of M if and only if M is a generic extension of N by some almost homogeneous notion of forcing. We also develop a slightly modified notion of FMS submodels to characterize the case where M is a generic extension of N not necessarily by an almost homogeneous notion of forcing.
"A Characterization of Permutation Models in Terms of Forcing." Notre Dame J. Formal Logic 43 (3) 157 - 168, 2002. https://doi.org/10.1305/ndjfl/1074290714