${\mathbb{C}\mathbb{P}}^2$-stable classification of $4$-manifolds with finite fundamental group

We show that two closed, connected $4$-manifolds with finite fundamental groups are ${\mathbb{C}\mathbb{P}}^2$-stably homeomorphic if and only if their quadratic $2$-types are stably isomorphic and their Kirby-Siebenmann invariant agrees.