normal subgroup and orbits

Let the group G act on the finite set A and let $\displaystyle H \trianglelefteq G$. Let $\displaystyle O_1, O_2,\ldots,O_r$ be the distinct orbits of H on A.

now read the following:

$\displaystyle g \cdot a_i = a_j$, where $\displaystyle g \in G, a_i \in O_i, a_j \in O_j$

let $\displaystyle b_i = h_1 \cdot a_i \text{ for some } h_1 \in H \text{ and } a_i \neq b_i$.

now $\displaystyle g \cdot b_i = g \cdot (h_1 \cdot a_i)=(g h_1) \cdot a_i = (g h_1 g^{-1} g) \cdot a_i = (g h_1 g^{-1})g \cdot a_i = h_2 g \cdot a_i = h_2 \cdot a_j \text{ where }g h_1 g^{-1} = h_2; h_2 \in H$.

now $\displaystyle h_2 \cdot a_j = c_j \text { for some } c_j \in O_j$. also it can be shown that $\displaystyle a_j \neq c_j$ very easily.

from the above argument can i say that ** $\displaystyle |O_i|=|O_j|$**