Check whether two conjugate subgroups are still conjugate in some subgroup

Question

Given a group $G$ and $K<G$. Let $H_1,H_2$ be subgroups in $G$ satisfying:

\begin{align} &H_1\triangleleft K;\\ &H_2\triangleleft K;\\ &H_1\sim_G H_2.\quad\mbox{(They are conjugate to each other in $G$)} \end{align}

Is it true that

\begin{align} H_1\sim_{N_G(K)}H_2, \end{align}

where $N_G(K)$ denotes the normalizer of $K$ in $G$? Or is there a counterexample?
Any idea would be appreciated.

Answer 1

$G=S_6$, $H_1=\langle (1,2,3) \rangle$, $H_2=\langle (4,5,6) \rangle$, $K = \langle (1,2,3),(4,5,6),(4,5) \rangle \cong C_3 \times S_3$.