Does there exist an abelian group that can be made into $\mathbb{Q}$-module in more than one way? Let $X$ and $Y$ denote $\mathbb{Z}$-modules. Then if $X$ and $Y$ have equal underlying abelian groups, we may deduce that $X=Y$.
Is this still true if we replace $\mathbb{Z}$ with $\mathbb{Q}$?
Equivalently, does there exist an abelian group that can be made into $\mathbb{Q}$-module in more than one way?

A $\mathbb Q$-module is a $\mathbb Q$-vector space.
Let $V$ be a $\mathbb Q$-vector space.
Then for each $n\in\mathbb N$ and $v\in V$, the vector $\frac 1nv$ is the unique(!) vector $w$ with $\underbrace{w+\ldots+w}_n=v$ and then $\frac mnv$ is of course uniquely determined as $\underbrace{w+\ldots+w}_m$ if $m\ge0$ and as the inverse of $\frac{|m|}nv$ if $m<0$.
