= Solution
Here transitivity is <order n-transitivity>: increasing tuples are compared, rather than arbitrary reorderings. For increasing tuples $\alpha_1<\cdots<\alpha_n$ and $\beta_1<\cdots<\beta_n$, choose $h_{ij}$ sending both $\alpha_i$ to $\beta_i$ and $\alpha_j$ to $\beta_j$ when $i\ne j$, using order two-transitivity. For $i=j$, choose an element sending the single point to its target, using transitivity. Define
$$
\boxed{h=\bigvee_{i=1}^n\bigwedge_{j=1}^nh_{ij}}.
$$
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-4-interpolation.png]
{title=Finite order interpolation constructed from pointwise minima and maxima}
This belongs to the <lattice-ordered permutation group>. At $\alpha_i$, all terms in its $i$th inner <meet> equal $\beta_i$. At $\alpha_k$, every inner <meet> is at most $\beta_k$, because its $j=k$ term takes that value. The outer <join> is consequently exactly $\beta_k$ there. Thus $h$ maps the whole tuple to its target. This <finite interpolation by lattice operations> proves every finite order-transitivity without assuming the group permits arbitrary cut-and-paste restrictions of its elements.
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