  In this paper we study families of representations of the outer automorphism
groups indexed on a collection of finite groups $\mathcal{U}$. We encode this
large amount of data into a convenient abelian category
$\mathcal{A}\mathcal{U}$ which generalizes the category of VI-modules appearing
in the representation theory of the finite general linear groups. Inspired by
work of Church--Ellenberg--Farb, we investigate for which choices of
$\mathcal{U}$ the abelian category is locally noetherian and deduce analogues
of central stability and representation stability results in this setting.
Finally, we show that some invariants coming from rational global homotopy
theory exhibit representation stability.
