Monodromy groups, i.e. the groups of isometries of the intersection lattice LX [colone] H2/torsion generated by the monodromy action of all deformation families of a given surface, have been computed by the author for any minimal elliptic surface with pg > q = 0. New and refined methods are now employed to address the cases of minimal elliptic surfaces with pg [ges ] q > 0. Thereby we get explicit families such that any isometry is in the group generated by their monodromies or does not respect the invariance of the canonical class or the spinor norm. The monodromy is also shown to act by the full symplectic group on the first homology modulo torsion.