Co-H-structures on equivariant Moore spaces

Let G be a finite group, ${\displaystyle O_G}$ the category of canonical orbits of G and ${\displaystyle A:O_G\to A}$b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with ${\displaystyle Ex{t}^{n-1}(A,A\otimes A)}$. Then the case ${\displaystyle G={\mathbb{Z}}_{p^k}}$ leads to an example of infinitely many G-homotopically distinct G-maps ${\displaystyle {\phi }_i:X\to Y}$ such that ${\displaystyle {\phi }_i^H}$, ${\displaystyle {\phi }_j^H:X^H\to Y^H}$ are homotopic for all i,j and all subgroups H ⊆ G.