What will be the orbit of the group action $G \times X \to X$ by $(g,H)=ghg^{-1}$ where $G$ is any group and $X$ is the set of all the subgroups of $G$?
If $H$ is normal then the orbit of $H$ will be $\{H\}$. But if $H$ is not normal then what will be the orbit of $H$? In case of $S_3$ I saw that the orbit of any subgroup of order $2$ is all the subgroups of order $2$. Is it true in general?
Yes, you are right about $S_3$, but it is not true in general. Consider, for instance $S_3\times\mathbb{Z}_2$ and take $H=\left\{(\operatorname{id},0),\bigl((1\ \ 2),0\bigr)\right\}$. Then $\{(\operatorname{id},0),(\operatorname{id},1)\}$ doesn't belong to the orbit of $H$, in spite of the fact that it is a subgroup with two elements.
In general, the orbit of $H$ will be a set of subgroups of $H$ all of which have the same order as $H$, but it doesn't have to be the set of all such subgroups.