In the theory of quantum automorphism groups, one constructs Hopf algebras acting on an algebra K from certain algebra morphisms σ:K→M _n(K). This approach is applied to the field K=k(t) of rational functions, and it is investigated when these actions restrict to actions on the coordinate ring B=k[t ²,t ³] of the cusp. An explicit example is described in detail and shown to define a new quantum homogeneous space structure on the cusp.