Let A be a pointed topological space. I want to show that A is a cogroup object in $Ho(Top_*)$ iff the functor $[A,\_] \colon Top_* \rightarrow Sets_*$ factors through the category of groups.
$\Leftarrow$: I want to find the multiplication and inverse map. Since $[A, A\vee A]$ is a group, it must contain an element (the identity). Let one of those representatives $m$ be the multiplication map.
One property the multiplication map must satisfy is for the composition $(*\vee id)\circ m$ to be homotopic to $id$. But already here I'm in trouble. The continuous function $* \vee id$ induces a group homomorphism $[A, A\vee A] \rightarrow [A,A]$. But since $m$ is the identity, this map must take $m$ to the identity of $A$. But it is not hard to see that the constant function $* \colon A \rightarrow A$ represents the identity in $[A,A]$. So if the property was true we would have a homotopy $id \sim *$. But this is only true when A is contractible.
Can someone help me finding the correct multiplication map (or point out my error above) and the correct inverse map (for the latter I only see two options, identity map or constant map).