  Given a self-similar groupoid action $(G,E)$ on the path space of a finite
graph, 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 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
${\mathcal G}(G,E)$ in certain cases and illustrate with an example.
