I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$).

Obvious candidates:

Lagrange resolvent (the reduction of quartic to cubic equations).

Tait's theorem on equivalence of 4-coloring of normal map and 3-coloring of its edges.

Do you have more interesting examples?