On smooth structures over $4$-manifolds with fundamental group of even order

We show that any topological, closed, oriented, non-spin $4$-manifold with fundamental group ${\mathbb{Z}}_{4k}$ and $\min (b_2^{+},b_2^{-})\ge 15$, has either none or infinitely many distinct smooth structures. Furthermore, we construct infinitely many non-diffeomorphic, irreducible, smooth structures on manifolds with signature zero, $b_2^{+}$ even and fundamental group ${\mathbb{Z}}_2\times G$, for any finite group $G$. This extends the results of Baykur-Stipsicz-Szabó.