 Equality-pattern orbits of ordered pairs

The diagonal group action of the symmetric group on ordered pairs has two orbits of a group action when $n\ge 2$: equal pairs, of size $n$, and distinct pairs, of size $n(n-1)$. Equality is preserved by a bijection, and a prescribed bijection on one or two distinct letters extends to a permutation of all letters. At $n=1$ only the equal-pair orbit remains.