Exotic definite four-manifolds with non-trivial fundamental group

Inspired by a recent result of Levine-Lidman-Piccirillo, we construct infinitely many exotic smooth structures on some closed four-manifolds with definite intersection form and fundamental group isomorphic to $\mathbb{Z}\mathrm{/}2\mathbb{Z}$. Similar constructions provide exotic smooth structures on further four-manifolds with fundamental group $Z\mathrm{/}2\mathbb{Z}$, including examples with even $b_2^{+}$.