Interesting problem. What have you tried?
Suppose $a,b \in x^{-1}Hx$. Can you show that $ab^{-1} \in x^{-1}Hx$?
(hint: first show that if $b = x^{-1}h'x$ that $b^{-1} = x^{-1}h'^{-1}x$ by multiplying the two together).
Next show that the map:
$h \mapsto x^{-1}hx$ is bijective between $H$ and $x^{-1}Hx$.
(it is fairly obvious it's surjective, so concentrate on proving it's injective).