The notion of semidirect product $\Gamma \rtimes G$ where $G$ is a group acting on a groupoid $\Gamma$ is set up in Chapter11, Section 11.4, of my book "Topology and groupoids"Groupoids".
It is used there in connection with studying orbit groupoids, and their relevance to the fundamental groupoid of an orbit space by a group action.
One nice point is that this semidirect product includes the case $\Gamma$ is a discrete groupoid, i.e. essentially a set, when you get what is commonly called the action groupoid. In this case the morphism $p: \Gamma \rtimes G \to G$ is known as a covering morphism of groupoids, and all covering morphisms of $G$ arise in this way.
I feel the use of covering morphisms of groupoids makes for a nice exposition, base point free, of the theory of covering spaces. Such an idea was pointed out for the simplicial case in the 1967 book on simplicial theory by Gabriel and Zisman, was used in the first 1968 edition of my book, and is used in Peter May's 1999 book "A concise course in algebrac topology".