 Solution

Letwith the lexicographic order, where the initial $\mathbb{Z}$ is one discrete block. This is a countable model of $\mathrm{Th}(\mathbb{Z},<)$: it is a discrete order without endpoints, and every interval is either of its prescribed finite length or contains arbitrarily long finite chains.

An element $a$ in the initial block and an element $b$ in a later block have the same one-type. There is, however, a $c<b$ with infinitely many points between $c$ and $b$, whereas no such $c<a$ exists because every predecessor of $a$ lies at finite distance within the initial block. The type of $(b,c)$ therefore cannot be transported over $a$, so $M$ is not aleph-zero-homogeneous.

Solved by gpt-5.6-sol high.