You may be thinking maps into a cogroup object should form a cogroup the way maps into a group object form a group. This isn't quite true; the "correct" analog is that maps out of a cogroup object form a group.
More precisely, if G is an object in a category and D is its group object diagram, then applying the functor Hom(X,-) to D yields a group object diagram making Hom(X,G) into a group. Essential here is the fact that Hom(X,-) preserves products (in fact all limits); it does not generally preserve coproducts, which is why in general one would not expect Hom(X,S) to inherit a cogroup structure from a cogroup structure on S. Hopefully this addresses your ponderings about the possibility of "cohomotopy groups".
The dual statement is true for cogroups: if S is a cogroup, applying Hom(_,XHom(-,X) to its cogroup diagram yields a diagram defining a group structure on Hom(S,X). Here, dually, it's essential that Hom(X,-Hom(-,X) turns coproducts into products (in fact all colimits into corresponding limits); it does not generally turn products in coproducts.
This makes precise the relationship of the cogroup structure on the spheres to homotopy groups: if you work through the example of Sn and see how its cogroup structure makes get on Hom(Sn,,X) into a group after applying Hom(-,X), I'm pretty sure it will coincide with you'll get exactly the nth homotopy group of X :)