## conjugacy in HNN extensions

Let $G=\langle t, A|t^{-1}Ht=K \rangle$ be an HNN extension with $H, K \in Z(A)$.
Let $x,y \in A$ such that $x \notin H \cup K$.
Then $x \nsim _{G} y$ if and only if $x \nsim_{A} y$.

It is obvious that if $x \nsim_{G} y$ then $x \nsim_{A} y$ since $A \subset G$.

I tried to prove the opposite direction but I have no idea where to use $H, K \in Z(A)$ in the proof.

Any hint??

Thank you.