Stallings' celebrated Fibration Theorem states that if a closed irreducible $3$-manifold $M$ admits a short exact sequence \begin{equation} 1 \to N \to \pi_1(M) \to \mathbb{Z} \to 1, \end{equation} where $N$ is finitely generated, then $M$ fibers over $S^1$.

My question is that whether it is possible to give the exact form of $M$ if we already know that $N$ is a surface group. More precisely, assume there is a short exact sequence \begin{equation} 1 \to \pi_1(S) \to \pi_1(M) \to \mathbb{Z} \to 1, \end{equation} where $S$ is a closed surface. Let $t \in \pi_1(M)$ be a preimage of $1 \in \mathbb{Z}$. Conjugation with $t$ induces an automorphism $\varphi$ of $\pi_1(S)$ whose projection $[\varphi] \in Out(\pi_1(S))$ in the outer automorphism group does not depend on the choice of $t$. By the Dehn-Nielsen-Baer Theorem the extended mapping class group $MCG^{\pm}(S)$ is isomorphic to $Out(\pi_1(S))$. Let $f:S \to S$ be a homeomorphism so that $[f] \in MCG^{\pm}(S)$ corresponds to $[\varphi] \in Out(\pi_1(S))$ under the Dehn-Nielsen-Baer isomorphism. My question is whether $M$ is diffeomorphic to the mapping torus $T_f$ of $f:S \to S$. If this is true, does it just follow from Stallings' proof? Or, in case it is known but requires more work, is there a good reference for it?

More concretely, I am interested in the following situation (which might simplify things). If $M$ is a closed hyperbolic $3$-manifold so that $\pi_1(M)$ is (abstractly) isomorphic to the fundamental group $\pi_1(T_f)$ of a mapping torus, then $M$ is diffeomorphic to $T_f$. This is, of course, an easy consequence of the Geometrization Conjecture. However, I would like to know whether there is a proof of this fact that does not depend on the solution of the Geometrization Conjecture nor on Thurston's mapping torus theorem.