We describe a system in which the self-replication of micellar aggregates results in a spontaneous amplification of chirality in the reaction products. In this system, amphiphiles are synthesized from two "clickable" fragments: a water-soluble "head" and a hydrophobic "tail". Under biphasic conditions, the reaction is autocatalytic, as aggregates facilitate the transfer of hydrophobic molecules to the aqueous phase. When chiral, partially enantioenriched surfactant heads are used, a strong nonlinear induction of chirality in the reaction products is observed. Preseeding the reaction mixture with an amphiphile of one chirality results in the amplification of this product and therefore information transfer between generations of self-replicating aggregates. Because our amphiphiles are capable of catalysis, information transfer, and self-assembly into bounded structures, they present a plausible model for prenucleic acid "lipid world" entities.