Determination of solvable algebraic groups by dense integral subgroups

Let G and G′ be triangularizable algebraic groups defined over the field Q of rational numbers, and let Γ ? G_Q and Γ′ ? G′_Q be dense subgroups of them containing integral subgroups of finite index. A study is made of the conditions under which a birational isomorphism of G and G′ follows from an abstract isomorphism of Γ and Γ′.