In my topology class, we tried to prove that pushouts of groups exists via the Seifert-van Kampen theorem. The main idea is that every group is isomorphic to a fundamental group of some path connected space (here, we used the presentation $\langle S \mid R \rangle$ and considered the mapping cone of a constructed map $\bigvee_{j \in R} S^1 \to \bigvee_{i \in I} S^1$).
So the idea is as follows: Given $G_1 \gets G_0 \to G_2$, we equivalently have $\pi_1(X_1, x_1) \gets \pi_1(X_0, x_0) \to \pi_1(X_2, x_2)$ whereby $G_k \cong \pi_1(X_k, x_k)$. Now we apply Seifert-van Kampen to obtain the pushout.
But in order to do so, we must construct some space $X$ such that $X_k \hookrightarrow X$, $X_1 \cup X_2 = X$ and $X_1 \cap X_2 = X_0$. How can this be done?
I think a good idea is for $X$ to be the some pushout of $X_1 \gets X_0 \to X_2$. But what should the maps be?