According to your definition, the separators will be exactly those groups G with a surjection to Z. One direction: if f(x)!=g(x), then take the composition G ->> Z -> A where the latter map sends the generator of Z to x. For the other direction, to distinguish the maps f,g: Z -> Z defined by f(x)=x and g(x)=2x, your separator must surject to Z.

This is a huge class of groups which has no particularly nice description beyond the definition, as far as I know.

According to your definition, the separators will be exactly those groups $G$ with a surjection to $\mathbb{Z}$. One direction: if $f(x)\neq g(x)$, then take the composition $G \twoheadrightarrow \mathbb{Z} \rightarrow A$ where the latter map sends the generator of $\mathbb{Z}$ to $x$. For the other direction, to distinguish the maps $f,g: \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $f(x)=x$ and $g(x)=2x$, your separator must surject to $\mathbb{Z}$.

This is a huge class of groups which has no particularly nice description beyond the definition, as far as I know.