Let $g$ be a positive integer, and let $G$ be a commutative group with the following constraint on its torsion subgroup: there is an injection $G[\operatorname{tors}] \hookrightarrow (\mathbb{Q}/\mathbb{Z})^{2g}$.  Must there be subgroups $G_1,\ldots,G_g$ of $G$ such that   

(i) $G = G_1 \times \ldots \times G_g$ (internal direct product), and  
(ii) For all $1 \leq i \leq g$, there is an injection $G_i[\operatorname{tors}] \hookrightarrow (\mathbb{Q}/\mathbb{Z})^2$?  

Motivation: If this is true, then it reduces the "Inverse Mordell-Weil Problem for Abelian Varieties" to the "Inverse Mordell-Weil Problem for Elliptic Curves".

Thus although the given question certain has an affirmative answer in many special cases -- e.g. it is a triviality if $G$ is finitely generated -- I am not really interested in that.  But it would be "lucky for me" if the answer turns out to be affirmative in the general case, so it's worth asking.