Topological classification of certain nonorientable 4-manifolds with cyclic fundamental group of order 2 mod 4

We show that the classification up to homeomorphism of closed topological nonorientable 4-manifolds with fundamental group of order 2 due to Hambleton-Kreck-Teichner can be used to classify a large set of such 4-manifolds with cyclic fundamental group of order 2p for every odd $p>1$. This is done through a simple cut-and-paste construction, and classical and modified surgery theory are used only through results of Hambleton-Kreck-Teichner and Khan. It is plausible that this set comprises all closed topological nonorientable 4-manifolds with ${\pi }_1=\mathbb{Z}\mathrm{/}2p$. We collect several interesting questions whose answers would guarantee a complete classification.