Here are two well-known facts:
Let $X$ be a topological space. We always have the diagonal map $\Delta :X\to X\times X$ and this induces a map $H^{\bullet}(X)\otimes H^{\bullet}(X) \simeq H^{\bullet}(X\times X )\to H^{\bullet}(X)$, which induces algebra structure on $H^{\bullet}(X)$. We call such multiplication as a cup product. (Let's forget about details about coefficients.)
Let $X$ be a topological group. Then the group operation map $X\times X\to X$ induces a comultiplication $H^{\bullet}(X) \to H^{\bullet}(X\times X)\simeq H^{\bullet}(X)\otimes H^{\bullet}(X)$ which is compatible with the cup product. Hence $H^{\bullet}(X)$ became a Hopf algebra.
Now here is my question: Let $X$ be a group, and we have a group operation map $X\times X\to X$ as before. Is it true that this induces a coalgebra structure on the group cohomology $H^{\bullet}(X,\mathbb{Z})\to H^{\bullet}(X, \mathbb{Z})\otimes H^{\bullet}(X, \mathbb{Z})$? If it is true, can we write down the map explicitly?