Let $\mathsf{hTop}_{\bullet}$ denote the homotopy category of pointed topological spaces. More precisely, the objects are pointed topological spaces and for two objects $X$ and $Y$, the morphisms from $X$ to $Y$ are the basepoint-preserving homotopy classes of maps from $X$ to $Y$ which are denoted by $[X, Y]_{\bullet}$.
A cogroup object in $\mathsf{hTop}_{\bullet}$ is an object $X$ such that for every object $Y$, $[X, Y]_{\bullet}$ is a group which is natural in $Y$. The positive dimensional spheres are examples of cogroup objects in $\mathsf{hTop}_{\bullet}$. More generally, the (reduced) suspension of any pointed topological space is a cogroup object in $\mathsf{hTop}_{\bullet}$.
According to nLab (see point 2), there are cogroup objects in $\mathsf{hTop}_{\bullet}$ (the page I link to writes $\mathsf{hTop}$, but I believe this is a typo) which are not suspensions, but no reference is given.
What is an example of a cogroup object in $\mathsf{hTop}_{\bullet}$ which is not a suspension?
If possible, I'd like to see a construction of such an object, but I'd also be satisfied with a reference.
I don't know much about this, but Berstein and Harper construct some such examples in this paper (which I haven't actually read). In particular, they construct an infinite family of 3-cell complexes which are cogroups but not suspensions, and also a 2-cell complex with one 5-cell and one 35-cell which is a cogroup but not a suspension. There is also a survey of this and related topics in Chapter 23 of the Handbook of Algebraic Topology (some of which you can read on Google books).
You may also be interested in the other answers I received when I asked a similar question many years ago on MathOverflow, though none of the answers say anything directly about how to construct an example.