Atkinson as well as Schönert and Seress describe methods to compute the minimal block system for transitive permutation groups; in particular in Permutation Group Algorithms by Ákos Seress, we find
Theorem 5.5.1 Suppose that a set S of generators for some transitive $G \leq Sym(\Omega)$ is given and $|\Omega| = n$. Then a minimal block of imprimitivity can be computed [...] by a deterministic algorithm.
Is there a way to compute such block systems for non-transitive permutation groups?
I have not found anything in the literature about such a computation.