Let $F$ be a torsionfree subgroup of a commutative group $G$. Are there nontrivial conditions known under which there exists a torsionfree direct summand of $G$ containing $F$?

I would already be happy with such a condition in case $F$ is free or $G$ is of finite type.

*Motivation:*

We consider a property $\mathcal{P}$ of graded rings and its behaviour under coarsening. Given an epimorphism of commutative groups $\psi:G\rightarrow H$ and a $G$-graded ring $R$ with $G$-graduation $(R_g)_{g\in G}$, the $\psi$-coarsening $R_{[\psi]}$ of $R$ is the $H$-graded ring with underlying ring the ring underlying $R$ and with $H$-graduation given by $(R_{[\psi]})_h=\bigoplus_{g\in\psi^{-1}(h)}R_g$ for $h\in H$.

Suppose we know that $\mathcal{P}$ is reflected by $\psi$-coarsening for every $\psi$, i.e., if $R_{[\psi]}$ has $\mathcal{P}$ then so does $R$, and that if $\ker(\psi)$ is a torsionfree direct summand of $G$ then $\mathcal{P}$ is respected by $\psi$-coarsening, i.e., if $R$ has $\mathcal{P}$ then so does $R_{[\psi]}$.

Then, we want to know whether $\mathcal{P}$ is also respected by $\psi$-coarsening if $\ker(\psi)$ is torsionfree but not necessarily a direct summand of $G$. And this would follow immediately if $\ker(\psi)$ is contained in a torsionfree direct summand of $G$.