  We study the orbits under the natural action of a permutation group $G
\subseteq S_n$ on the powerset $\mathscr{P}(\{1, \dots , n\})$. The permutation
groups having exactly $n+1$ orbits on the powerset can be characterized as
set-transitive groups and were fully classified in \cite{BP55}. In this paper,
we establish a general method that allows one to classify the permutation
groups with $n+r$ set-orbits for a given $r$, and apply it to integers $2 \leq
r \leq 15$ using the computer algebra system GAP.
