This question already has an answer here:
Let $G,H$ be groups. Suppose we have both an action of $G$ on $H$, and an action of $H$ on $G$, both non-trivial. Let "$\cdot$" define the former action, and $\circ$ define the latter.
What can we say about the product: $$ (g,h)\,(g',h') := \big(g\,(h\circ g'), h\,(g\cdot h')\big), $$
or the product: $$ (g,h)\,(g',h') := \big(g\,(h\circ g'), (g'\cdot h)\,h'\big), $$
or anything similar where the other action is a right action?
Do we have a group? Is this used somewhere? Is it useful, or too general?