G is a finite group that acts transitively on $X$. And H is a normal subgroup of G. The question asks about the size of orbit under the induced action of $H$ on $X$. I pick up $x$, $y$ from set $X$ and write it as $gx=y$.
Then I established the following: $g$$H_{x}$$g^{-1}$=$H_{y}$
($x$ and $y$ may not on the same orbit of action $H$, but I still prove that they have conjugate stabilizers)
Then I try to use the Orbit-Stabilizer Theorem which indicate the bijection between the orbit of $x$ under $H$, which is $Ox$ and the left cosets of the stabilizer $h$$H_{x}$.
Then |$Ox$|= |$h$$H_{x}$| and |$Oy$|= |$h$$H_{y}$|
I feel that might be something wrong cause I just stuck here. Are there any way that I could establish $h$$H_{x}$$h^{-1}$=$H_{y}$ instead of $g$$H_{x}$$g^{-1}$=$H_{y}$?
Thanks!