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 $\Gamma\subset G_Q$ and $\Gamma'\subset G'_Q$ 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 Gamma and Gammaprime.