  For a group $G$, $\mathcal{F}_G$ denotes the set of all non-empty finite
subsets of $G$. We extend the finitary coarse structure of $G$ from $G\times G$
to $\mathcal{F}_G\times \mathcal{F}_G$ and say that a macro-uniform mapping $f:
\mathcal{F}_G \rightarrow \mathcal{F}_G$ (resp. $f: [G]^2 \rightarrow G$) is a
finitary selector (resp. 2-selector) of $G$ if $f(A)\in A$ for each $A\in
\mathcal{F}_G$ (resp. $ A \in [G]^2 $). We prove that a group $G$ admits a
finitary selector iff $G$ admits a 2-selector and iff $G$ is a finite extension
of an infinite cyclic subgroup or $G$ is countable and locally finite. We use
this result to characterize groups admitting linear orders compatible with
finitary coarse structures.
