A counterexample on cyclic blocks in finitary permutation groups

We give a negative answer to Kourovka Notebook Problem 21.6.  Starting from a minimally transitive permutation \(2\)-group of degree eight with an explicit cycle-support obstruction, we form a restricted iterated imprimitive wreath limit.  The result is a countable transitive totally imprimitive finitary \(2\)-group containing no transitive subgroup with the requested cyclic-block property.

This manuscript addresses the intended mathematical problem identified in the source.

AI disclosure: This problem was solved using GPT 5.6 Sol.

Mathematics subject: math.GR; source problem: OP-0695.