Consider the following combinatorial problem:
Let $G$ be a finite group, and $X = \sqcup_{i\in I} X_i$ be a finite set. Suppose that for each $g\in G$ and $i\in I$ we have sets $Y_{g,i} \subset X$, which are each unions of some of the $X_i$.
Define a constrained action $\rho$ of $G$ on $X$ to be an action $\rho: G \to \operatorname{Aut}(X)$ such that $\rho(g)(x) \in Y_{g,i}$ for every $g \in G$ and $x \in X_i$.
The permutations $\prod_i S_{X_i}$ act on such constrained actions: if $\pi \in \prod_i S_{X_i}$, then with $\rho^{\pi}(g)(x) = \pi^{-1} \rho(g)(\pi x)$, we see that $\rho^{\pi}$ is again a constrained action.
Find representatives of each $\prod_i S_{X_i}$ orbit of constrained actions.
I can think of a number of approaches to this; so far they are either too dumb to work in the actual instances of this problem I care about, or too complicated for me to want to implement yet.
Does anyone know an off-the-shelf (e.g. implemented in an existing CAS) algorithm for this problem?