A morphism $i : A \to B$ has the left lifting properties with respect to a morphism $p : X \to Y$ if and only if any commutative square with $i$ on the left and $p$ on the right has a diagonal lift making everything commute:
It is easy to see that the canonical inclusion of any group $G$ in a free product $G \to G * H$ has the left lifting property with respect to all morphisms with a right inverse. It made me think about the converse: is it true that for a morphism $f : G \to H$, there exists a group $G'$ and an isomorphism $\phi : H \to G * G'$ such that $\phi \circ f$ is the canonical inclusion if and only if $f$ has the left lifting property with respect to all morphisms with right inverse?
An interesting corollary, if this proposition is true, is that for a morphism $f : G \to H$, there is a free group $F$ and an isomorphism $\phi : H \to G * F$ such that $\phi \circ f$ is the canonical inclusion if and only if it has the left lifting property with respect to all surjective morphisms.