Consider a set $X$ with an action of the group $G$, and a set $Y$ with an action of the group $H$.
Consider now a function $f:X\to Y$. It makes no sense to talk about equivariance, since $G$ and $H$ are in general different. However we have an action of $G\times H$ on $Y^X$ (set of functions from $X$ to $Y$).
What can we say about this action? Is there a standard theory to study its behavior?
In particular, what if there are $g\in G,h\in H$ such that $fg = hf$?
Now suppose that $f$ is a bijection between $X$ and $Y$. This way $G$ can act on $Y$ (conjugating by $f$) and $H$ can act on $X$. What can we say? For example, what would be the kernel and the stabilizers of the $G\times H$ action in this case? Have things like these beens studied?
Any reference would also be welcome.