I have a general-type question: Suppose $M$ is a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an automorphism of $M$. Also, let $Aut(M)$ be the group of automorphisms of $M$ equipped with the pointwise convergence topology.

I want to know necessary and/or sufficient conditions under which $Aut(M)$ preserves a linear ordering on $\mathbb{N}$, in the sense that $Aut(M)$ preserves $(\mathbb{N},\prec)$ iff for every $n,m\in\mathbb{N}$ and for every $g\in Aut(M)$, $$n\prec m\Leftrightarrow g(n)\prec g(m).$$