The second paragraph of this chapter informally describes a homomorphism from Sn to Z2, where the even permutations all map to one of the elements and the odd permutations all map to the other element. Replace Sn by S6 and replace Z2 by the permutation version of the cyclic subgroup of order 2, and construct a nontrivial homomorphism between these two groups. Evaluate your homomorphism with enough even and odd permutations to be convinced that it is correct. Then construct the kernel and verify that it is the group you expect.
Hints: First, decide which elements of the group of order 2 will be associated with even permutations and which will be associated with odd permutations. Then examine the generators of S6 to help decide just how to build the homomorphism.