I know that if a pointed topological space $(Q,\ast)$ is an $H$-cogroup, then we can define a functor: $$[(Q,\ast),-]:\mathbf{Top}_\ast\to \mathbf{Groups}$$ that acts similarly to the homotopy groups functors $\pi_n$. Clearly $\pi_n=[(S^n,\ast),-]$. However I didn't find interesting examples or usages of such functors when $Q$ is not a sphere.
Are spheres the only interesting $H$-cogroups?
All reduced suspensions $S(X,x_0)$ are $h$-cogroups. This allows to give all stable homotopy sets $\{(X,x_0) , (Y,y_0)\} = \varinjlim [(S^n(X,x_0),S^n(Y,y_0)]$ a natural structure of an abelian group.
There are also $h$-cogroups which are no suspensions, but they are fairly exotic. See Example of a cogroup in $\mathsf{hTop}_{\bullet}$ which is not a suspension.