let $hTop_*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone give an explicit example, with proof? I know that homotopy colimits are related to this, but they don't seem to be categorical colimits, so I don't think that they fit here.

especially I'm interested in the following special case: let $G= \langle X | R \rangle$ a presentation of a group and consider the resulting map $\omega : \vee_{r \in R} S^1 \to \vee_{x \in X} S^1$. does the cokernel of $\omega$ exist in $hTop_*$? in $Top_*$, the cokernel is just the 2-dimensional CW-complex $Q$, which is optained from $\vee_{x \in X} S^1$ via the attaching map $\omega$. now if $f : \vee_{x \in X} S^1 \to T$ is a pointed map such that $f \omega$ is nullhomotopic, it is easy to see that it extends to a map $\overline{f} : Q \to T$. but I think that we cannot expect that $\overline{f}, \overline{g}$ are homotopic, when $f,g$ are homotopic: the homotopies between $f$ and $g$ don't have to be compatible. can you give an example for that? probably it already works for $\omega : S^1 \to S^1, z \mapsto z^2$, thus $Q = \mathbb{R} P^2$.

anyway, this only would show that $Q$ is not the cokernel in the category $hTop_*$. the proof, that the cokernel does not exist at all, will be even more difficult and I don't know how to approach it.

you may also replace the category by $hCW_*$ (CW-complexes), $hCG_*$ (compacty generated spaces) etc., if it's useful.