  We study representations of wreath product analogues of categories of finite
sets. This includes the category of finite sets and injections (studied by
Church, Ellenberg, and Farb) and the opposite of the category of finite sets
and surjections (studied by the authors in previous work). We prove noetherian
properties for the injective version when the group in question is
polycyclic-by-finite and use it to deduce general twisted homological stability
results for such wreath products and indicate some applications to
representation stability. We introduce a new class of formal languages
(quasi-ordered languages) and use them to deduce strong rationality properties
of Hilbert series of representations for the surjective version when the group
is finite.
