The fundamental 2-group of a pointed space is the monoidal groupoid with weak inverses given by the ‘reverse path’ map. This description was given in Carrasco-Cegarra-??. There was an equivalent definition given in HDA V: 2-groups, as the automorphism 2-group of the point , considered as an object of the fundamental bigroupoid.
The fundamental 2-group plays a role in the theory of 2-covering spaces analogous to that of that of the fundamental group in the theory of covering spaces, but the theory is not all worked out yet. See chapters 4 and 5 of my thesis.