Let $X$ a affine algebraic variety. Then $X$ can have infinite automorphism group, for example $X=\mathbb{A}^1$. Let $G$ be a divisible abelian group. My question is about some condition in $G$ such that I can guarantee 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 orbit of $P$ by $G_p$ is rational over $\mathbb{Z}_p$. I want to know if there is some bibliography about this problem.