Injectivity of maps in a free product group with amalgamation

Question

Given countable groups $H, G_1$ and $G_2$ and group homomorphisms $\theta_i: H \rightarrow G_i$, we can construct the free group with amalgamation $(G_1 \ast G_2)/N$ where $N$ is the normal subgroup generated by the elements $\theta_1(h) \theta_2(h^{-1})$ with $h \in H$. We have canonical homomorphisms $\rho_i: G_i \rightarrow (G_1 \ast G_2)/N$.

My question is: if $\theta_1$ is injective, does it follow that $\rho_2$ is injective? References are welcome as well!

The question has its origin from category theory, where the amalgamated free product is the pushout. In some categories my question has a positive answer.