Invariant factors of a subgroup of a subgroup of $\mathbb{Z}^2$

Question

Consider groups $B\leq A\leq\mathbb{Z}^2$.

We have:

1. A basis $e_1,e_2$ of $\mathbb{Z}^2$ and integers $a_1,a_2$ such that $a_1e_1,a_2e_2$ is a basis for $A$, and $a_1\mid a_2$.

2. A basis $f_1,f_2$ of $A$ and integers $b_1,b_2$ such that $b_1f_1,b_2f_2$ is a basis for $B$, and $b_1\mid b_2$.

3. A basis $g_1,g_2$ of $\mathbb{Z}^2$ and integers $c_1,c_2$ such that $c_1g_1,c_2g_2$ is a basis for $B$, and $c_1\mid c_2$.

Given $a_1,a_2,b_1,b_2$, what are all the possibilities for $c_1,c_2$?