By using large and small convex $\ell$-subgroups of an $\ell$-group introduced in [3] and [4], we extend $\ell$-isomorphisms of $\ell$-groups to quasi-$\ell$-isomorphisms and co-quasi-$\ell$-isomorphisms. Then we will establish some sufficient and necessary conditions under which two $\ell$-groups are quasi-$\ell$-isomorphic or co-quasi-$\ell$-isomorphic.