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An group automorphism preserves element orders and sizes of conjugacy classes. A nonidentity involution in $S_n$ has cycle type $2^k{1}^{n-2k}$ and class sizeFor its class to have the same size as the transposition class, it must satisfyFor fixed $k\ge 2$ the left side grows strictly with $n$. When $k=2$, it is $(n-2)(n-3)$, which is two at $n=4$ and at least six at $n=5$, never the required four. For $k=3$ equality first occurs at $n=6$, since $4!=2^2\cdot 3!$, and thereafter the left side is larger. For $k\ge 4$ it is already larger at $n=2k$: the ratio $(2k-2)!/[2^{k-1}k!]$ is $15/4$ at $k=4$, and increases with $k$, its successive ratio being $k(2k-1)/(k+1)>1$. This proves the symmetric-group involution class-size collision result. Therefore, for $n\ne 6$, every group automorphism maps transpositions to transpositions.

Now recover the underlying points from transpositions. Two distinct transpositions fail to commute exactly when their two-element supports meet. A maximal family of pairwise intersecting edges of a complete graph is either a vertex star or a triangle. Indeed, after selecting edges $\{1,2\}$ and $\{1,3\}$, every edge either contains 1 or is $\{2,3\}$; if the latter occurs, all edges lie in that triangle. For $n\ge 5$, the stars have size $n-1\ge 4$ and triangles size three, so stars are recognized purely by the multiplication structure of $S_n$.

An group automorphism must thus permute the stars, giving a permutation $\tau$ of the points. The unique transposition shared by stars $i,j$ is $(ij)$, so its image is $(\tau (i)\tau (j))$. Since transpositions generate $S_n$, the whole group automorphism is conjugation by $\tau$.

At $n=4$, stars and triangles both have three elements, but their generated subgroups distinguish them: the transpositions in a star generate all of $S_4$ (since $(ij)=(ai)(aj)(ai)$ for its center $a$), while a triangle generates the $S_3$ fixing the fourth point. The same reconstruction follows. At $n=3$, conjugation by $S_3$ induces every permutation of its three transpositions (a transposition swaps the other two, and a three-cycle rotates all three), so a transposition-preserving group automorphism is again inner. At $n=2$ the only group automorphism of $C_2$ is the identity, and $n=0,1$ are trivial (if the empty-set degree is included). ThusThe small-degree adjustments are necessary: a cardinality-only star argument would not cover $n=4$.