I have a question concerning this article (https://math.berkeley.edu/~qchu/TQFT.pdf) about TQFTs and representation theory of finite groups.
In the beginning of the third section, it is stated that, for a finite group $G$, a $G$-cover is the same as a groupoid homomorphism $$\Pi_1(M)\to BG,$$ between the fundamental groupoid of a manifold $M$ and the groupoid consisting of one object whose automorphisms are the elements of $G$.
I would like to get my head around this correspondence, but I don't know much about principal $G$-bundles, so I would appreciate any hints or reference recommendations that would allow me to understand it in further detail.
This is known as the classification of coverings. It is often given as a statement about the fundamental group, but can be easily generalized to fundamental groupoids.
A detailed exposition in the language of groupoids is available in Ronald Brown's Topology and Groupoids.