The locally quaternion group is a countable $B$-group

A group $G$ is a $B$-group if every primitive permutation group containing a regular subgroup isomorphic to $G$ is doubly transitive. We prove that the locally quaternion group $\mathrm{Dic}(C_{2^\infty},c)$, the extension of the Prüfer $2$-group by an element squaring to its involution and invertin