Given a countable group G, the space of all measure-preserving actions of G is naturally endowed with a Polish space structure; it is then interesting to understand which properties are generic. I’ll talk about what is known on this subject, and focus on a related question: assume that H is a subgroup of a countable group G; when is it true that, for any comeager set of measure-preserving H-actions, the set of G-actions whose restriction belongs to this prescribed set is also comeager?