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 non-trivial 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.