Let \(G\) be a group. If \(\Omega\) and \(\Delta\) are (right) \(G\)-sets, a function \(\phi : \Omega \to \Delta\) is called an almost \(G\)-map if, for all \(g \in G\), the set \(\{\omega \in \Omega \mid (\phi \omega) g \neq \phi (\omega g)\}\) is finite. Let \(\{*\}\) denote the \(G\)-set consisting of a single \(G\)-fixed element, \(*\). If \(G\) is free on a set \(S\), then the Cayley graph for \(G\) with respect to \(S\) is a right \(G\)-tree \(T\) with a base vertex \(v_0 = 1\), and \(T\) is \(G\)-free, in the sense that point stabilizers are trivial. There is a bijective map \(\phi\) from the vertex set, \(G\), to the set \((S \times G) \vee \{*\}\) which consists of the disjoint union of \(\{*\}\) and the edge set, \(S \times G\); the map \(\phi\) assigns to a vertex \(v\) the first edge in the geodesic path in \(T\) from \(v\) to the base vertex \(v_0\), where this edge is taken to be \(*\) in the case where \(v = v_0\). It is readily verified that the bijective map \(\phi : G \to (S \times G) \vee \{*\}\) is an almost \(G\)-map. This fact was observed and exploited in the theory of von Neumann algebras by M. Pimsner, D. Voiculescu, J. Cuntz, P. Julg, A. Valette, A. Connes and P. A. Linnell. In this paper, it is shown that \(G\) is a free group if and only if there exist \(G\)-free \(G\)-sets \(\Omega\) and \(\Delta\), and a bijective almost \(G\)-map \(\phi : \Omega \to \Delta \vee \{*\}\); this answers a question raised orally by \textit{P. A. Linnell}, who was motivated by the fact that the latter property is the only one he uses when dealing with free groups in his article [Forum Math. 5, 561-576 (1993; Zbl 0794.22008)]. The paper under review goes on to give necessary and sufficient conditions on the rank of a free group \(G\) and the number of orbits in \(G\)-free \(G\)-sets \(\Omega\) and \(\Delta\) for there to exist a bijective almost \(G\)-map \(\phi : \Omega \to \Delta \vee \{*\}\). For example, there exists a bijective almost \(G\)-map \(\phi : G \to G \vee \{*\}\) if and only if \(G\) is either free of rank 1 or free of infinite rank.