Let $K \le G$ be a subgroup. I have to show that if $F$ is a free $\mathbb{Z}G$-module, then it is also free as a $\mathbb{Z}K$-module.
The definition I have in mind is that $F=\left\{ \sum r_x x \mid r_x \in \mathbb{Z}G, x \in X \right\}$ for some set $X$ where each of the sums is finite.
At least from this definition, it isn't apparent at all that we can restrict the $r_x$ to $\mathbb{Z}K$ and get the same result. How should I proceed from here?