Let $\prod_{n=1}^{\infty}\mathbb{Z}$ be the Baer-Specker group and $\bigoplus_{n=1}^{\infty}\mathbb{Z}$ be the natural free abelian subgroup. It is known that if $G$ is a countable abelian group with no infinitely divisible elements (e.g. $\mathbb{Z}$), then every homomorphism $\prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z}\to G$ is trivial.

I've heard by word of mouth of a result on fundamental groups which would imply the existence of non-trivial homomorphisms $\prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ for any prime $p\geq 2$. What is an explicit construction of such a homomorphism for given $p$?