  Only two examples of extremally Ricci pinched G2-structures can be found in
the literature and they are both homogeneous. We study in this paper the
existence and structure of such very special closed G2-structures on Lie
groups. Strong structural conditions on the Lie algebra are proved to hold. As
an application, we obtain three new examples of extremally Ricci pinched
G2-structures and that they are all necessarily steady Laplacian solitons. The
deformation and rigidity of such structures are also studied.
