We classify n-point extensions of ergodic Z^d-actions up to relative orbit equivalence and establish criteria under which one n-point extension of an ergodic Z^d-action can be sped up to be relatively isomorphic to an n-point extension of another ergodic Z^d-action. Both results are characterized in terms of an algebraic object associated to each n-point extension which is a conjugacy class of subgroups of the symmetric group on n elements.