Permutation Groups and Set-Orbits on the Power Set

A permutation group G acting on a set Omega induces a permutation group on the power set P(Omega) (the set of all subsets of Omega). Let G be a finite permutation group of degree n and s(G) denote the number of orbits of G on P(Omega). It is an interesting problem to determine the lower bound inf (log(2) s(G)/n) over all groups G that do not contain any alternating group A(l) (where l > t for some fixed t >= 4) as a composition factor. The second author obtained the answer for the case t = 4 in Yang (J Algebra Appl 19:2150005, 2020). In this paper, we continue this investigation and study the cases when t >= 5, and give the explicit lower bounds inf (log(2) s(G)/n) for each positive integer 5 <= t <= 166.