Lie group actions induces a manifold structure

Question

Let $G$ be a Lie group, and let $H$ be the underlying group structure of $G$.

Let $A$ be a set equipped with an group action (say left) $\rho:H\times A\to A$.

Does $A$ heiress any structure of a manifold from $G$?

Thanks in advance.

Answer 1

Every finite or countable set is naturally a $0$-dimensional manifold, but that has nothing to do with $G$. Other than that, there is nothing you can say. Suppose, for instance, that $G=(\Bbb R,+)$, that $A=\Bbb Q$ and that$$(\forall x\in\Bbb R)(\forall q\in\Bbb Q):\rho(x,q)=0.$$Then, as above, the only Lie group structure you can put on $A$ is to see it as a $0$-dimensional manifold.