Let $G$ be a locally compact abelian group. A *lattice* in $G$ is a discrete subgroup $\Lambda$ such that the quotient $G / \Lambda$ is compact. A *Borel fundamental domain* of a lattice $\Lambda$ in $G$ is a Borel set $F \subseteq G$ that intersects each coset in $G / \Lambda$ in exactly one point, i.e., $F$ is a set of coset representatives.

My question is now the following:

Given two lattices $\Lambda_1, \Lambda_2 \subseteq G$ such that $\Lambda_1 \subseteq \Lambda 2$, is it always possible to take corresponding fundamental domains $F_i$ of $\Lambda_i$, where $i = 1,2$, such that $F_2 \subseteq F_1$?

By looking at special cases in, say, $G=\mathbb{R}$ or $G = \mathbb{Z}$, it seems that this is always possible.

Any comment or reference is highly appreciated.