Let \(G\) be a permutation group on a countably infinite set \(\Omega\), and for every positive integer \(k\) let \(n_ k\) denote the number of orbits under the action of \(G\) on subsets of order \(k\) of \(\Omega\). It was proved by \textit{P. J. Cameron} [Math. Z. 148, 127-139 (1976; Zbl 0313.20022)] that the sequence \((n_ k)_{k\in \mathbb{N}}\) is nondecreasing. Moreover, \textit{P. J. Cameron} started [in J. Lond. Math. Soc., II. Ser. 17, 410-414 (1978; Zbl 0385.20002)] the study of permutation groups with \(1 < n_ k = n_{k+1} < \infty\) for some \(k\). Here the author considers transitive groups with \(1 < n_ 2 = n_ 3 < \infty\) and transitive imprimitive groups for which \(1 < n_ 4 = n_ 5 < \infty\).