Is the kernel of a map from a $\mathbb{Z}$-free module to an abelian group $\mathbb{Z}$-free?

Question

Is the kernel of a map from a $\mathbb{Z}$-free module to an abelian group $\mathbb{Z}$-free? It seems like the answer should be yes as the kernel is contained inside the free $\mathbb{Z}$-module but I can't quite put my finger on the precise reason why.