 Freeness detected by a quotient

Suppose a group homomorphism $\pi :G\to Q$ is injective on a subgroup freely generated by $x_1,\dots ,x_n$. If $y_i$ have $\pi (y_i)=\pi (x_i)$, then the $y_i$ freely generate their subgroup: a relation among them projects to a relation among the free basis images, so the corresponding abstract reduced word must be empty. In particular, this applies to $y_i=x_{i}g$ when the normal closure of $g$ intersects $⟨x_1,\dots ,x_n⟩$ trivially.