Homotopy classification of $4$-manifolds with $3$-manifold fundamental group

We give a criterion on a group $π$ and a homomorphism $w \colon π\to C_2$ under which closed $4$-manifolds with fundamental group $π$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic $2$-types. We verify the criterion for a large class of $3$-manifold groups and orientation characters, in particular for the fundamental group $π$ of any closed, orientable $3$-manifold whose finite subgroups are cyclic, provided $w$ vanishes on every element of $π$ of finite order. We deduce a homeomorphism classification of closed, orientable $4$-manifolds with infinite dihedral fundamental group $\mathbb{Z}/2 * \mathbb{Z}/2$.