Infinite imprimitive non abelian group?

Question

My new question is

Is there an infinite, imprimitive and non abelian group?

Thank you for the further answers.

Answer 1

Consider the subgroups $A={\rm Sym}(2{\bf Z})$ and $B={\rm Sym}(1+2{\bf Z})$ sitting inside $G={\rm Sym}({\bf Z})$. So $G$ is the set of bijections from the set of integers to itself, $A$ is the set of permutations of the even integers or equivalently the permutations which fix all odd integers, and $B$ the set of permutations of the odd integers or equivalently the permutations which fix all even integers. Let $h:x\mapsto x+1$ be the simple forward translation map. Then $H=\langle A,B,h\rangle$ is an infinite group which acts transitively on the integers and preserves the nontrivial partition ${\bf Z}=2{\bf Z}\cup(1+2{\bf Z})$.