  We give a complete classification of homomorphisms from the commutator
subgroup of the braid group on $n$ strands to the braid group on $n$ strands
when $n$ is at least 7. In particular, we show that each nontrivial
homomorphism extends to an automorphism of the braid group on $n$ strands. This
answers four questions of Vladimir Lin. Our main new tool is the theory of
totally symmetric sets.
