Group action, so that every subgroup is a stabilizer?

Question

In any group action, stabilizers are subgroups of the group.

Question: Given a finite group $G$, does there always exists a set $X$ and an action of $G$ on $X$ such that every subgroup of $G$ is stabilizer of some $x\in X$?

Answer 1

Yes, one easy example is $G$ acting by left multiplication on the disjoint union $$\large X=\bigsqcup_{\substack{\text{subgroups}\\ H\subseteq G}} G/H$$ Clearly $H$ is the stabilizer of the coset $eH\in G/H$, and since $X$ includes a copy of every $G/H$, every subgroup of $G$ occurs as a stabilizer at least once.

This also doesn't require $G$ to be finite.

Answer 2

Yes. Let $G$ act on the power set of its underlying set by elementwise left multiplication. (This does not require $G$ to be finite)