Find number of normal subgroups in $F_3$ such that its factor is isomorphic to a given Abelian group https://ift.tt/eA8V8J

On the upcoming test I will be given a problem of type:

Find all normal subgroups $H$ in $F_n$ such that $F_n/H \cong G$.

Here $n$ is a small integer, likely 2 or 3, and $G$ is an Abelian group given as product of some cyclic groups.

Question: is there a more-or-less routine algorithm for such problems?

I do know an approach. I count two parameters: the number of epimorphisms $\#(F_n \twoheadrightarrow G)$ and the number of automorphisms $\#Aut(G)$. The answer then is $\frac{\#(F_n\twoheadrightarrow G)}{\#Aut(G)}$. This solution is based on the following facts: 1) every normal subgroup is a kernel of some homomorphism; 2) $F_n/Ker\,\phi \cong G$ if $\phi$ is an epimorphism; 3) $Ker\,\phi=Ker\,\psi$ iff $\psi = \alpha\circ\phi$ for some $\alpha \in Aut(G)$.

The problem is, very often I'm not sure how do I count one or both of these parameters. I could go through all homomorphisms and automorphisms manually, but the numbers in problem are usually too large.

Here are some cases to show how big numbers might be on the actual test. No need to solve them!

1. $n=3$, $G=\mathbb Z_{13}^{3}$
2. $n=3$, $G=\mathbb Z_{10}^3\times \mathbb Z$
3. $n=3$, $G=\mathbb Z_{70}\times \mathbb Z_{30} \times \mathbb Z_{15}$
4. $n=2$, $G=\mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5$
5. $n=3$, $G=\mathbb Z_{55} \times \mathbb Z_{7} \times \mathbb Z_{77}$
6. $n=3$, $G=\mathbb Z_{75} \times \mathbb Z_{375} \times \mathbb Z_{125} \times \mathbb Z_{333}$

from Hot Weekly Questions - Mathematics Stack Exchange
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