Suppose that $G= \mathbb{Z}^n \times H$ for some group $H$. Then, there is a short exact sequence $$ 1\rightarrow \mathbb{Z}^n \rightarrow G \rightarrow H \rightarrow 1$$ which right splits (clearly).
Let $K< G$ be a subgroup. Then, the previous short exact sequence descends to another SES $$ 1 \rightarrow \mathbb{Z}^n \cap K \rightarrow K \rightarrow p(K) \rightarrow 1.$$
Does this short exact sequence right split? If $x\in p(K)$, then there is an element $z x \in K$ for some $z\in \mathbb{Z}^n$. So I could define $\iota \colon p(K) \mapsto K$ such that $\iota(x)= zx$. However, is $\iota$ a homomorphism?
It does not necessarily split, because $K$ can be a nontrivial subdirect product.
As an example, take $n=1$, $H = {\mathbb Z}$, where the normal subgroup and $H$ are generated by $x$ and $y$, respectively, and let $K$ be the subgroup $\langle x^2,y^2,xy \rangle$.