A permutation group G acting on a set Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} induces a permutation group on the power set P(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {P}}(\Omega )$$\end{document} (the set of all subsets of Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}). Let G be a finite permutation group of degree n and s(G) denote the number of orbits of G on P(Ω)\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {P}}(\Omega )$$\end{document}. It is an interesting problem to determine the lower bound inflog2s(G)n\documentclass[12pt]{minimal}
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\begin{document}$$\inf \left( \frac{\log _2 s(G)}{n}\right) $$\end{document} over all groups G that do not contain any alternating group Aℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{A}_\ell $$\end{document} (where ℓ>t\documentclass[12pt]{minimal}
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\begin{document}$$\ell >t$$\end{document} for some fixed t⩾4)\documentclass[12pt]{minimal}
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\begin{document}$$t\geqslant 4)$$\end{document} as a composition factor. The second author obtained the answer for the case t=4\documentclass[12pt]{minimal}
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\begin{document}$$t=4$$\end{document} in Yang (J Algebra Appl 19:2150005, 2020). In this paper, we continue this investigation and study the cases when t⩾5\documentclass[12pt]{minimal}
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\begin{document}$$t\geqslant 5$$\end{document}, and give the explicit lower bounds inflog2s(G)n\documentclass[12pt]{minimal}
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\begin{document}$$\inf \left( \frac{\log _2 s(G)}{n}\right) $$\end{document} for each positive integer 5⩽t⩽166\documentclass[12pt]{minimal}
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\begin{document}$$5\leqslant t\leqslant 166$$\end{document}.
