Fix a prime $p$ and let $x$ be a $p$-adic integer that is not an integer. Now let $M$ be the group of pairs of rational numbers $(r,s)$ so that $rx-s$ is a $p$-adic integer. Then $M$ maps onto $\mathbb Q$ but it does not contain a copy of $\mathbb Q$, because that would mean that some $s/r$ would not just be a good approximation of $x$, but a perfect one.

Fix a prime $p$ and let $x$ be a $p$-adic integer that is not a rational number. Now let $M$ be the group of pairs of rational numbers $(r,s)$ so that $rx-s$ is a $p$-adic integer. Then $M$ maps onto $\mathbb Q$ but it does not contain a copy of $\mathbb Q$, because that would mean that some $s/r$ would not just be a good approximation of $x$, but a perfect one.

Fix a prime $p$ and let $x$ be a $p$-adic integer that is not an integer. Now let $M$ be the group of pairs of rational numbers $(r,s)$ so that $rx-s$ is a $p$-adic integer. Then $M$ maps onto $\mathbb Q$ but it does not contain a copy of $\mathbb Q$, because that would mean that some $r/s$ would not just be a good approximation of $x$, but a perfect one.

Fix a prime $p$ and let $x$ be a $p$-adic integer that is not an integer. Now let $M$ be the group of pairs of rational numbers $(r,s)$ so that $rx-s$ is a $p$-adic integer. Then $M$ maps onto $\mathbb Q$ but it does not contain a copy of $\mathbb Q$, because that would mean that some $s/r$ would not just be a good approximation of $x$, but a perfect one.