A free action of a group G on a space X leads to a bijection of X x G onto the graph of the action in the fiber product of X with itself over the space of orbits Y. The Galois isomorphism is the k-dual, and the k-valued function algebra on X is a Hopf-Galois extension of the function algebra on Y, with the k-dual of G being the Hopf algebra coacting. Relaxing the bimodule isomorphism condition of Hopf-Galois extension to split epimorphism captures the notion of depth two from subfactors. Depth two algebra extensions are Galois where bialgebroids and Hopf algebroids coact. Computing their "Galois groupoids" is to find new types of Hopf algebroids in quantum algebra.