 Solution

A permutation preserves whether the two coordinates are equal. Consequently the diagonal and its complement are invariant under this group action. Any $(i,i)$ can be carried to any $(j,j)$ by a permutation sending $i$ to $j$. Likewise, for $i\ne j$ and $k\ne l$, the assignment $i\mapsto k$, $j\mapsto l$ extends to a permutation of the remaining letters.

For $n\ge 2$ the equality-pattern orbits of ordered pairs are the two orbits of a group action are thereforewith sizes $n$ and $n(n-1)$. For $n=1$ only the diagonal orbit of a group action exists.