Let $G$ be a group and put \begin{align} \mathrm{FIA}(G)&=\{N\triangleleft G:G/N\textrm{ is finite abelian}\}. \end{align} Question: Are there some general ways to compute $\bigcap\mathrm{FIA}(G)$? What if, especially, $G$ is a free group?
My motivation is to get the universal covering of a (nice) topological space among finite abelian coverings, in the sense that the group above is the corresponding fundamental group.
For example, if $G$ is a finite abelian group, then $\mathrm{FIA}(G)=\{e\}$: the trivial group. Same for the case $G=\mathbb{Z}$. Another example is $\mathrm{FIA}(S_n)=A_n$ for $n\geq 5$. I have no idea for $G=F_2$ or other free groups.
Thank you!