I think I roughly understand the quantum algorithm for the general Abelian Hidden Subgroup Problem (HSP). We begin by constructing a uniform superposition, calculating the function over that superposition in another register, then measure it to yield a coset state (i.e. a state whose nonzero components all lie in the same coset). Applying the Quantum Fourier Transform will then yield an element that is "orthogonal," or in the kernel, of the hidden subgroup. Repeating this only polynomially many times, we can be certain with arbitrarily high probability that we have produced the hidden subgroup.
My question is, where does this go wrong for non-Abelian subgroups? And what exactly about being Abelian is so important for this algorithm to work correctly? If someone could provide an intuitive explanation, that would be even better! Thanks.