Let $G_1$ and $G_2$ be countable abelian groups, and let $\iota\colon G_1\to G_2$ be an injective group homomorphism (so that we may regard $G_1$ as a subgroup of $G_2$). Suppose that for every finitely generated subgroup $K$ of $G_2$, there is a map $\pi_K\colon K\to G_1$ such that $\iota \circ \pi_K\circ \iota= \mbox{id}_{\iota(G_1)\cap K}$ (or if you want to omit $\iota$ from the notation, this would be $\pi_K(g)=g$ whenever $g\in G_1\cap K \subseteq G_2$.

In other words, for every finitely generated subgroup of $G_2$, one can find a splitting. Does it follow that there is a "global" splitting $\pi\colon G_2\to G_1$ for $\iota$?

I don't want to assume that the "partial" splittings $\pi_K$ have any sort of coherence (by this I mean that I don't want to assume that whenever $K\subseteq K'$, the splitting $\pi_{K'}$ can be chosen so that $\pi_{K'}$ restricted to $K$ is just $\pi_K$). I also don't want to assume that $G_1$ is finitely generated.

Of course the conclusion would be that $G_1$ is a direct summand in $G_2$.