Consider a semidirect product of $G_{a,b}={\mathbb R}\ltimes {\mathbb R}^2$. The action is diagonal with two eigenvalues $a$ and $b$. Its Lie group isomorphism class is determined by $a/b$. 

Not let $\phi :  {\mathbb R}\rightarrow {\mathbb R}$ be an isomorphism of ${\mathbb Q}$-vector spaces. Then $G_{a,b}$ is obviously isomorphic to $G_{\phi(a),\phi(b)}$ as abstract groups but $\phi(a)/\phi(b)$ could be easily different from $a/b$.