Let $G$ be a group and $X$ a $G$-set. Is there a name already used to describe the following scenario?
The stabilizer of each point of $X$ in $G$ is either trivial or the whole group.
Hence, the orbit space decomposes as just a bunch of copies of $G$ and singletons, no in-between. The name "$0$-$1$ action" is lying on my subconscious but I don't know why and I don't recall where I might have seen it. Any references are very much appreciated!
On the points whose stabiliser is the whole group, $G$ acts trivially. On the points whose stabiliser is trivial, $G$ acts freely. There is no need for a name for the behaviour across all of $X$ because the action decomposes.