Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order.

Furthermore, suppose that the complement of $A$ in $\mathbb{Q}$ is not finite.

- (@) Does there exist an order-automorphism $\alpha$ of

$(\mathbb{Q},\leq)$ such that $A \subset \alpha(A) \ne A$ ?

(The condition on $\vert \mathbb{Q} \setminus A \vert$ might be too naive, but I am looking for "as mild as possible" conditions on $A$ so that (@) would be true.)