Let $H$ be a bialgebra and $A$ a $H$-module algebra. The cross product $A \rtimes H$ is defined as follows. As a vector space $A \rtimes H = A \otimes H$. The multiplication on $A \rtimes H$ is defined as follows: \begin{align} (a \otimes h)(b \otimes g) = \sum a(h_{(1)}.b) \otimes h_{(2)}g, \end{align} where $a, b \in A$, $h, g \in H$, $\Delta(h) = \sum h_{(1)} \otimes h_{(2)}$.

Is there a comultiplication $\Delta$ on $A \rtimes H$ such that the cross product $A \rtimes H$ is a bialgebra? If $\Delta$ does not always exist, under what conditions there is a comultiplication $\Delta$ on $A \rtimes H$ such that $A \rtimes H$ is a bialgebra? Thank you very much.

Edit: we add one more condition: suppose that $A$ is a bialgebra. In this case, is there a comultiplication $\Delta$ on $A \rtimes H$ such that the cross product $A \rtimes H$ is a bialgebra? If $\Delta$ does not always exist, under what conditions there is a comultiplication $\Delta$ on $A \rtimes H$ such that $A \rtimes H$ is a bialgebra? Thank you very much.