We consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤ
m
, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤ
m
, 2) and of the homotopy classes of maps from M(ℤ
m
, 2) to M(ℤ
n
, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤ
m
, 2) to M(ℤ
n
, 2) for each co-H-structure on M(ℤ
m
, 2) and on M(ℤ
n
, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤ
n
, 2) on the set of co-H-structures of M(ℤ
m
, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤ
m
, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤ
m
, 2) are associative and non-commutative.