Given a Lie group \(G\) the authors consider the co-adjoint representation \(G\to Aut(\mathcal G^*)\) of the group \(G\) in the algebra of linear operators on the dual space \(\mathcal G^*\) to the Lie algebra \(\mathcal G\) of \(G\). For certain classes of 5-dimensional Lie groups \(G\) (called MD5-groups) the authors describe the geometric structure of orbits of the co-adjoint representation of \(G\). This paper can be considered as a continuation of the paper [Le Anh Vu, East-West J. Math. 7, No. 1, 13–22 (2005; Zbl 1120.22005)] devoted to the classification of MD5-groups.