Let $(\Omega,\leq)$ be a countable linear order. Suppose that for every finite $m \in \mathbb{N}$, and all subsets $S_1$ and $S_2$ of $\Omega$ of order $m$, there is an order-automorphism of $(\Omega,\leq)$ that sends $S_1$ to $S_2$.

Call the order-automorphism group $A$.

Is there a "fundamental theorem" for $(\Omega,\leq)$ in the spirit of the *fundamental theorem of projective geometry*, of the following form:

- if $\Omega_1$ and $\Omega_2$ are subsets of $\Omega$, order-isomorphic through an order-isomorphism $\gamma$, then there is an order-automorphism $\overline{\gamma}$ in $A$ which extends $\gamma$.

Of course, this statement is too general -- there should at least be some restrictions on $\Omega_i$ -- but I am looking for a statement "as general as possible."

Second related question: what if we remove the countability condition on $\Omega$ ?