When is the following map possible?

$A\boxtimes A(\mathcal{G}\times\mathcal{G})\rightarrow A(\mathcal{G}) \otimes A(\mathcal{G})$; where $\mathcal{G}$ is a group scheme, $A$ is a quasi coherent sheaf(of algebras) over $\mathcal{G}$ and $\boxtimes$ is the external tensor product of a sheaf given by $\pi_1^*A\otimes\pi_2^* A $.

In general, for any open set $U$, $A\boxtimes A(U\times U)\rightarrow A(U)\otimes A(U)$ doesnt hold true. What are the conditions required for this to hold? Will generation of $A$ by global sections suffice?

Please help.