Let $X$ a affine algebraic variety. We have that $X$ can have infinite isomorphisms, for example let $X=\mathbb{A}^1$. Let $G$ be a divisible group.
My question is about some condition in $G$ such that I can garantize the existence of a algebraic variety $X$ such that $G$ is a subgroup of the isomorphism group of $X$. 

More specifically let $G_p$ the $p$-primary component of $G$. I want to construct $X$ with a rational point $P$ such that the orbite of $P$ by $G_p$ is rational over $\mathbb{Z}_p$. I want to know if there is some bibliography about it problem?.