  In this article we develop a notion of soficity for actions of countable
groups on sets. We show two equivalent perspectives, several natural properties
and examples. Notable examples include arbitrary actions of both amenable
groups and free groups, and actions of sofic groups with locally finite
stabilizers. As applications we prove soficity for generalized wreath products
(and amalgamated free generalized wreath products) of sofic groups where the
underlying group action is sofic. This generalizes the result of Hayes and Sale
[HS18], and proves soficity for many new families of groups.
