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Given a self-similar groupoid action $(G,E)$ on the path space of a finitegraph, we study the associated Exel-Pardo \'etale groupoid ${\mathcal G}(G,E)$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 aboutsimilarity of groupoids which resembles Takai duality. We also describe ageneral strategy to compute the $K$-theory of $C^*(G,E)$ and the homology of${\mathcal G}(G,E)$ in certain cases and illustrate with an example.

On groupoids and $C^*$-algebras from self-similar actions