Let $G$ be a (countable) group, $H<G$ be a proper subgroup and $\theta:H\to G$ be an injective group homomorphism such that $\theta(H)\neq G$. Consider the HNN extension $\Gamma=<G,t \mid tHt^{-1}=\theta(H)>$

Similarly to link, if $H$ and $\theta(H)$ are malnormal in $\Gamma$, then the centralizer of a nontrivial element in $\Gamma$ is either infinite cyclic or it is in a conjugate of $G$.

Is it sufficient to assume that $H$ and $\theta(H)$ are malnormal in $G$ ? (eventually replacing cyclic by virtually cyclic ?)

Let $G$ be a (countable) group, $H<G$ be a proper subgroup and $\theta:H\to G$ be an injective group homomorphism such that $\theta(H)\neq G$. Consider the HNN extension $\Gamma=<G,t \mid tHt^{-1}=\theta(H)>$

Similarly to link, if $H$ and $\theta(H)$ are malnormal in $\Gamma$, then the centralizer of a nontrivial element in $\Gamma$ is either infinite cyclic or it is in a conjugate of $G$.

Is it sufficient to assume that $H$ and $\theta(H)$ are malnormal in $G$ ? (eventually replacing cyclic by virtually cyclic ?)