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An automorphism preserving the transposition class is inner: commuting relations recover the vertex stars of the complete graph, with a small-case adjustment in degrees three and four. An involution class has the same size as the transposition class only in degree six, when triple transpositions also have size fifteen. Hence outer automorphism groups of symmetric groups are trivial outside degree six and have size at most two in degree six.

// Target: combinatorics.bigb

Fix a basepoint $x_0$. A homeomorphism $f:S\to S$ induces an isomorphism from ${\pi }_1(S,x_0)$ to ${\pi }_1(S,f(x_0))$. Choosing a path from $x_0$ to $f(x_0)$ identifies the latter group with $G={\pi }_1(S,x_0)$. Changing the path conjugates the resulting automorphism by an element of $G$, while an isotopy changes nothing in the outer automorphism group. Hence there is a natural homomorphismthe mapping-class action on the outer automorphism group of the fundamental group.