Intersection of orbits is the orbit of intersection

Question

Let $G$ be a (topological) group acting on a space $X$. Let $x\in X$ and $A,B$ be two subgroups of $G$. Is it true in general that $(A\cdot x)\cap (B\cdot x)= (A\cap B)\cdot x$?

Answer 1

Let $G = C_2 \times C_2$ and $X = \Bbb Z/2\Bbb Z$ and $(g^a,g^b) \cdot \overline n := \overline{a + b + n}$ and $A = \{(e,g), (e,e)\}$ and $B = \{(g,e), (e,e)\}$.

Then, LHS = $X$ and RHS = $\{\overline 0\}$.