Consider $A := \mathrm{Aut}(\mathbb{Q},\leq)$, the group of order-automorphisms of $(\mathbb{Q},\leq)$. Call a subgroup $U$ highly order-transitive if for any two finite ordered sequences $s_1$ and $s_2$ of the same size, there is an order-automorphism of $U$ which sends $s_1$ to $s_2$, and note that $A$ has this property.
It is well known that $A$ contains countable subgroups which are also highly order-transitive (whereas $A$ is not countable). For instance, the free groups $F_\eta$ with $\eta \geq 2$ and finite can be embedded in $A$ with this property.
My question is: do there exist countable, highly order-transitive subgroups $U$ of $A$ with the following additional property:
- There exists an element $u$ in $U$ which fixes every element of $\mathbb{Q}$ outside some open interval $(a,b)$ with $a \ne b$ and $a, b \in \mathbb{Q}$, and no element in $(a,b)$ itself.
