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First determine when a group automorphism can carry a transposition to another type of involution. An involution in $S_n$ is a product of $k$ disjoint transpositions, and its conjugacy class has sizeAn automorphism preserves element orders and conjugacy-class sizes. Equality with the transposition-class size $n(n-1)/2$ requiresFor $k=2$ this says $(n-2)(n-3)=4$, which has no integral solution $n\ge 4$. For $k=3$, the minimum value of the left side, at $n=6$, is $4!=24=2^2\cdot 3!$; it increases strictly for larger $n$. For $k\ge 4$, even its minimum $(2k-2)!$ exceeds $2^{k-1}k!$: at $k=4$ this is $720>192$, and the ratio of these two quantities increases at the next step by $(2k)(2k-1)/(2(k+1))>1$. Thus only in degree six can the transposition class be exchanged with a different involution class, namely the triple transpositions.

We next prove that any automorphism preserving transpositions is an inner automorphism. Regard a transposition as an edge of the complete graph on the $n$ letters. Two distinct transpositions fail to commute exactly when their edges meet. Any family of pairwise intersecting edges is contained in a vertex star or a triangle: after choosing $\{a,b\}$ and $\{a,c\}$, an edge not containing $a$ must be $\{b,c\}$, which precludes any fourth edge. For $n\ge 5$, the stars are therefore distinguished as the maximal such families of size $n-1\ge 4$. An automorphism permutes stars, inducing a permutation $\pi$ of the letters; their pairwise intersections show that $(ij)$ is sent to $(\pi (i)\pi (j))$. Since transpositions generate $S_n$, this is conjugation by a permutation.

For $n=4$, stars and triangles both have three edges, but a star generates $S_4$ while a triangle generates a subgroup $S_3$. An automorphism preserves the order of the generated subgroup and hence still distinguishes stars. For $n=3$, every permutation of the three transpositions is induced by conjugation in $S_3$, since its conjugation action on them is faithful and has order six. The groups $S_1,S_2$ have trivial automorphism groups. This completes the small cases.

It follows that all automorphisms for $n\ne 6$ are inner. In degree six, an automorphism permutes the two involution classes of size 15; the kernel of this action consists precisely of the inner automorphisms. Therefore the automorphisms of the symmetric group satisfy