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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 :)
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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 objects will 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,_Hom(X,-) to D yields a group object diagram making Hom(X,G) into a group(since covariant hom . Essential here is the fact that Hom(X,-) preserves products (in fact all limits); it does not generally preserve coproducts)coproducts, which is why in general one would not expect Hom(X,S) to inherit a cogroup structure from a cogroup structure on S.

The dual statement is true for cogroups: if H S is a cogroup, applying Hom(_,X) to its cogroup diagram yields a diagram defining a group structure on Hom(H,XHom(S,X). Here, dually, it's essential that Hom(X,-) (since contravariant hom turns coproducts into products (in fact all colimits into corresponding limits); it does not generally preserve products)turn products in coproducts.

It may be interesting

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; , I'm pretty sure it will coincide with the nth homotopy group of X , but I haven't checked this myself.:)
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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 analog is that maps out of a cogroup objects will 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 (since covariant hom preserves products; it does not generally preserve coproducts).

The dual statement is true for cogroups: if H is a cogroup, applying Hom(_,X) to its cogroup diagram yields a diagram defining a group structure on Hom(H,X) (since contravariant hom turns coproducts into products; it does not generally preserve products).

It may be interesting to work through the example of Sn and see how its cogroup structure makes get on Hom(Sn,,X) into a group; I'm pretty sure it will coincide with the nth homotopy group of X, but I haven't checked this myself.