\mathbb Z$.

As $H$ is finitely generated and $\left.p\right|_H$ is not trivial and not injective, by Proposition \[rewrite\], there is an isomorphism $\iota: F_m*_A \to H$, natural fibration $\pi: F_m*_A \to \mathbb Z$, and injective map $n:\mathbb Z \to \mathbb Z$ such that $n(1) \ge 1$ and $\left.p\right|_H \iota = n \pi$, where $F_m*_A$ is an HNN extension of a finitely generated free group $F_m$ over a free factor $A \le F_m$. As $\chi(H) = 0$, $A$ is not a proper free factor by Lemma \[hwlem\]. Therefore, $H \cong F_m*_{F_m}$. As $F_m$ is finitely generated, there is an $i \ge 0$ such that $K = \iota(F_m) \le t^i F t^{-i} \le \ker p$. Fix large enough $i$, then $K \le t^i F t^{-i}$ is a finitely generated nontrivial subgroup such that $i_x\bar\phi^n(K) \le K$, where $n = n(1)$, $\bar \phi$ is the natural extension of $\phi$ to $t^i F t^{-i}$, and $x \in t^i F t^{-i}$. As $\phi$ is irreducible and atoroidal, so is $\bar \phi$. By Proposition \[invSbgrp\], $(i_x \bar\phi^n)^{-k}(K)$ has finite index in $t^i F t^{-i}$ for some $k \ge 0$. Therefore, $H = \langle (t^n x)^k K (t^n x)^{-k}, t^n x \rangle$ has finite index in $G 