Permutation groups with few orbits on the power set. II

We continue the study of permutation groups acting on the power set $\mathscr{P}(\{ 1, 2, \dots, n \})$. Permutation groups must have a minimum of $n + 1$ set-orbits. Previously in [3], the authors of that paper used GAP to classify permutation groups with a low number of orbits for permutation groups having $n + r$ set-orbits for some given $2 \leq r \leq 15$. We develop improvements to their theory and algorithms in GAP to classify further cases, from $16 \leq r \leq 33$.