If $\phi:G\twoheadrightarrow H$ and if $H=H_1*_{H_3}H_2$ for $H_i\leq H$, then is $G=G_1*_{G_3}G_2$, for $\phi(G_i)=H_i$?

Question

If $G$ is a group, with epimorphism $\phi \colon G\rightarrow H$, and if $H=H_1*_{H_3}H_2$ for $H_i\leq H$, then is it true that $G=G_1*_{G_3}G_2$, where $\phi(G_i)=H_i$? If yes, how? If not, what would be a counterexample?

I tried to universal property of the amalgamated free product, but failed.