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

We give a criterion on a group $\pi$ and a homomorphism $w \colon \pi \to C_2$ under which closed $4$-manifolds with fundamental group $\pi$ 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 $\pi$ of any closed, orientable $3$-manifold whose finite subgroups are cyclic, provided $w$ vanishes on every element of $\pi$ of finite order. We deduce a homeomorphism classification of closed, orientable $4$-manifolds with infinite dihedral fundamental group $\mathbb{Z}/2 * \mathbb{Z}/2$.