Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \setminus L$? If so, is there any way one can distinguish the constructible sets of Vitali's type from the nonconstructible sets of Vitali's type?

By a set of Vitali's type it is meant a subset of $\mathbb{R}$ containing exactly one element of every equivalence class of the relation $x - y \in G$, where $G$ is some fixed countable subgroup of $( \mathbb{R} , + )$.

Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \backslash L$? If so, is there any way one can distinguish the constructible sets of Vitali's type from the nonconstructible sets of Vitali's type?

By a set of Vitali's type it is meant a subset of $\mathbb{R}$ containing exactly one element of every equivalence class of the relation $x - y \in G$, where $G$ is some fixed countable subgroup of $( \mathbb{R} , + )$.

Consider sets of Vitali's type in models of $ZF+GCH$ where $L$ $\neq$$V$. Are there sets of Vitali's type in both $L$ and $V-L$? If so, is there any way one can distinguish the constructible sets of Vitali's type from the nonconstructible Vitali sets?

Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \setminus L$? If so, is there any way one can distinguish the constructible sets of Vitali's type from the nonconstructible sets of Vitali's type?

By a set of Vitali's type it is meant a subset of $\mathbb{R}$ containing exactly one element of every equivalence class of the relation $x - y \in G$, where $G$ is some fixed countable subgroup of $( \mathbb{R} , + )$.