If $\Gamma$ has a subgroup $\Gamma_1$ of index 2, so $\Gamma = \Gamma_1 \cup (\Gamma_1 + r)$ for any $r \in \Gamma \backslash \Gamma_1$, and $V$ is a Vitali set with respect to $\Gamma$, then $V \cup (V + r)$ is a Vitali set with respect to $\Gamma_1$. There are dense subgroups of $\mathbb R$ that have subgroups of index 2, e.g. ${\mathbb Q} + (2 \alpha) {\mathbb Z}$ is a subgroup of index 2 of ${\mathbb Q} + \alpha {\mathbb Z}$ where $\alpha$ is irrational.

If $\Gamma$ has a subgroup $\Gamma_1$ of index 2, so $\Gamma = \Gamma_1 \cup (\Gamma_1 + r)$ for any $r \in \Gamma \backslash \Gamma_1$, and $V$ is a Vitali set with respect to $\Gamma$, then $V \cup (V + r)$ is a Vitali set with respect to $\Gamma_1$. There are dense subgroups of $\mathbb R$ that have subgroups of index 2, e.g. $\alpha {\mathbb Z} + (2 \beta) {\mathbb Z}$ is a subgroup of index 2 of $\alpha {\mathbb Z} + \beta {\mathbb Z}$ where $\alpha$ and $\beta$ are linearly independent over $\mathbb Q$.