Tensor product of groups has a crossed module structure with respect to each group

Statement
Suppose $$G$$ and $$H$$ are groups with a compatible pair of actions on each other. Consider the tensor product of groups $$G \times H$$. Then, $$G \otimes H$$ has a natural crossed module structure over $$G$$. Also, $$G \otimes H$$ has a natural crossed module structure over $$H$$.

The crossed module structure over $$G$$ is defined as follows:


 * The group action of $$G$$ on $$G \otimes H$$ arises from group acts naturally on its tensor product with any group.
 * The homomorphism from $$G \otimes H$$ to $$G$$ arises from tensor product of groups maps to both groups.

The crossed module structure over $$H$$ is defined as follows:


 * The group action of $$H$$ on $$G \otimes H$$ arises from group acts naturally on its tensor product with any group.
 * The homomorphism from $$G \otimes H$$ to $$H$$ arises from tensor product of groups maps to both groups.

The fact that the axioms for crossed modules are satisfied follows (through some tedious computation) from the axioms for a compatible pair of actions.