If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that the theory of a countable structure $\mathcal{M}$ is $\aleph_0$ categorical if and only if the action of $\text{Aut}\mathcal{M}$ on $M$ is oligomorphic.
So if we start with an $\aleph_0$ categorical theory, the automorphism group of the countable model gives an oligomorphic action. Conversely, if a group $G$ acts oligomorphically on a set $X$, then the theory of $X$ with an $n$-ary predicate for each orbit of $X^n$ is $\aleph_0$-categorical.
Is there any structure of a category on groups with an action and on $\aleph_0$-categorical theories that can make this correspondence functorial? (For example, two bi-interpretable theories should give rise to isomorphic group actions, and the reduct of a theory should give a richer group)