Let $F_2$ be the free group on two generators.

Let $U\le F_2$ be a characteristic subgroup of finite index, and let $f : F_2\rightarrow\mathbb{Z}^2$ be the abelianization map.

It's easy to check that $f(U)$ is characteristic in $\mathbb{Z}^2$, so it must be of the form $N\mathbb{Z}\times N\mathbb{Z}$ for some $N\ge 1$, and hence the induced quotient $\mathbb{Z}^2/f(U)$ must be of the form $(\mathbb{Z}/N\mathbb{Z})^2$.

I apologize if this is a basic question, but my general question is - Is there a way of determining the $N$ from the characteristic subgroup $U$?

Of course this is too vague - I suppose an underlying question was - what are the characteristic subgroups of $F_2$? (There seems to be very little literature on this)

**A more precise question might be:**

Let $G$ be a finite group, and let $H\lhd F_2$ with $F_2/H \cong G$, and let $\{H_i\}$ be the $Aut(F_2)$-orbit $H$, then $U_H := \bigcap_i H_i$ is characteristic in $F_2$. Then we know that $f(U) = N\mathbb{Z}\times N\mathbb{Z}$. Is there a way of determining $N$ from $H$?

If we let $U_G$ be the intersection of *all* $G$-defining subgroups of $F_2$, then is there a way of determinine the $N$ associated to $f(U_G)$ from $G$?