I have two sets $X_1$, $X_2$ each with a corresponding group action $G_1$, $G_2$. Linking the two sets is $f:X_1\to X_2$ that maps orbits of $G_1$ into the same point in $X_2$. In other words $X_1/G_1 \simeq f(X_1)$. Thus, we can think of the group $G_2$ as acting on the orbits $X_1/G_1$, which gives rise to object I'm trying to understand, the quotient $(X_1/G_1)/G_2$. Is this an object commonly studied? What type of structure does it have?
The main question I care about is: Under what conditions there exist a group $H$ acting on $X_1$ s.t. $X_1/H \simeq (X_1/G_1)/G_2$?
An example: $X_1=X_2=\mathbb R^2$. The first group is the 4 reflections along the axis $G_1\simeq C_2\times C_2$. $f$ is just the component wise absolute value, mapping the whole plane into the positive quadrant. The second group $G_2$ is any line subset under addition (the kernel of a linear map $A:\mathbb R^2 \to \mathbb R$).
