Let f : G → H be a fixed homomorphism and p′ : G * H → G and p″ : G * H → H the two projections of the free product. Then a co-action relative to f is a homomorphism s : G → G * H such that p′s = id and p″s = f. We study this notion and investigate the following questions. What restrictions does s place on the structure of the group G? What form does s take in special cases? When does s induce a co-multiplication on H? What is the relation between associativity of s and associativity of the induced co-multiplication m on H? What are the properties of the operation of Hom(H, B) on Hom(G, B) induced by s : G → G * H? In addition, we give several diverse examples of co-actions in the last section.