What do you call subsets of $G$-sets whose elements are permuted under some action $G$?

Question

Given any group $G$ acting on any set $X$ via some left or right action $\varphi:G\times X\to X$ is there a name for subsets $Y\subseteq X$ with the property that for any $g\in G$ wge have $\{\varphi(g,y):y\in Y\}=Y$?

So that intuitively any element of $G$ acts on the elements of $Y$ by permuting them.

Answer 1

"G-invariant" is mentioned in the comments above, though technically you should say "G-invariant subset" since other things than subsets (e.g., properties) can also be G-invariant. The other, even simpler, name is "a union of orbits".