Green relations over finite monoids of $G$-equivariant functions

For a group $G$ acting over a set $X$, the set of all the $G$-equivariant functions, i.e., the set of functions which conmute with the action, ($g\cdot f(x)=g\cdot f(x),\mathrm{\forall }g\in G,\mathrm{\forall }x\in X$), is a monoid with the composition. The Green Relations are powerful tools to comprehend the structure of a semigroup. We study the case where $X$ is a finite set and compute the green relations for its monoid of $G$-equivariant functions, attempting to describe them based on some particular elements in the monoid called elementary collapsings.