As an example to set the scene, suppose I have a cube and I let its symmetry group act on its faces. Every face admits four transformations which leave the cube invariant and which maps the chosen face onto itself. The orbit of each face consists of six elements since there are six possible positions for a face to end. By the orbit-stabilizer theorem, the symmetry group of the faces of a cube therefore contains 24 elements.
In this case, it is possible to construct the symmetry group from the action in the following way. Fix a neutral position of the cube and let $F$ be the face on top in this neutral position. Now take another position of the cube where $F^\prime$ is the face on top. It is possible to find an element $g$ in $G$ which maps $F$ to $F^\prime$ and hence $g^{-1}$ maps $F^\prime$ to $F$. Now we can find an element $h$ in the stabilizer of $F$ such that $hg^{-1}$ maps the cube back to its neutral position. This shows that the symmetry group of the faces of a cube is generated by the stabilizer of any face of the cube along with a set of elements in the group which allows the face to reach its full orbit.
My question is now if it is possible to generalize this procedure to arbitrary finite groups. More precisely, let $G$ be a finite group acting on a (finite) set $X$ and let $x$ be an arbitrary element of $X$. Is $G$ generated by the stabilizer of $x$ along with elements of $G$ which map $x$ to each element of its orbit?