A proper permutation group transitive on countable sections

Let \\(\\Omega\\) be a countably infinite set. We prove that a proper transitive subgroup of \\(\\mathrm{Sym}(\\Omega)\\) acts transitively on partitions of \\(\\Omega\\) into countably many infinite parts, answering Kourovka Problem 9.41(c) without an additional set-theoretic hypothesis. After identifying \\(\\Omega\\) with \\(\\mathbb{N}\\), we use the group of permutations that are increasing on the parts of a finite partition. Every prescribed matching of two countable partitions is realized by a permutation increasing on two complementary subsets. The proof applies the relational Schröder–Bernstein theorem to increasing embeddings that preserve the part labels.