Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank submodule, so that $\mathbb{Z}^n/\Lambda$ is a finite abelian group. Assume that we know a set of generators of $\mathbb{Z}^n/\Lambda$. Can we determine generators of the larger group $\mathbb{Z}^n/(\ell\cdot\Lambda)$ for $\ell \in \mathbb{N}$ in terms of generators of $\mathbb{Z}^n/\Lambda$?
EDIT: In my case, $\mathbb{Z}^n/\Lambda \cong \mathbb{Z}/(2) \times \mathbb{Z}/(2)$ and I know two elements $x_1,x_2 \in \mathbb{Z}^n$, s.t. $\langle[x_1],[x_2]\rangle =\mathbb{Z}^n/\Lambda$. Do the cosets $x_i+\ell\Lambda$ generate $\mathbb{Z}^n/(\ell\cdot\Lambda)$?
I have the feeling that $\mathbb{Z}^n/\ell\Lambda \cong \mathbb{Z}/(\ell)^{n-2} \times \mathbb{Z}/(2\ell)^2$ in this case and so I need the other $n-2$ basis vectors for the generatig system..
Consider the canonical map $\Bbb Z^n/(\ell\cdot\Lambda) \to \Bbb Z^n/\Lambda$, and take the full preimage of the given generators.
This will be certainly a generator system, of size multiplied by $\ell$.
(However, I guess, this is far from optimal, i.e. one could sift this set to obtain a smaller generator set.)