Given a self-similar groupoid action (G, E) on the path space of a finite graph, we study the associated Exel-Pardo étale groupoid g(G, E) and its C∗ -algebra C∗ (G, E). We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the K-theory of C∗ (G, E) and the homology of g(G, E) in certain cases and illustrate with an example.