Given a group G acting both on the left and the right of a set X, say the actions are compatible if $g \cdot (x \cdot g') = (g\cdot x)\cdot g'$ for all $g,g'\in G$ and $x\in X$. 


Here are some examples of compatible $G$-actions:


1) For any left action (resp. right action), that action and the trivial right action (resp left action) on the same set are compatible. 


2) For any group G, its actions on itself by left and right multiplication are compatible (this is just associativity of the group law).


3) For any commutative group G, take any left or right action on a set X and let it act by the same map to Aut(X) on the other side, and by commutativity these actions are compatible.


We can also compose these examples with products and surjective group maps to obtain more compatible actions.


My question is, are there other naturally arising compatible actions?



Sets with compatible right and left G-actions
Suppose we have a group $G$ acting on the right and on the left as above on a set $X$. These actions give rise to a left action of $G\times G$ on $X$ by setting $(g\_1,g\_2)\cdot x=g\_1\cdot x \cdot g\_2^{-1}$. Conversely, if we have a left action of $G\times G$ on a set $X$, we can construct compatible left and right actions by setting $g\cdot x= (g,e)\cdot x$ and $x\cdot g= (e,g^{-1})\cdot x$. The compatibility follows from the fact that the factors of $G\times G$ commute.


Of course, instead of left actions of $G\times G$ one can also consider right actions.


