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There are two very natural products of compact matrix quantum groups: thetensor product $G\times H$ and the free product $G*H$. We define a number offurther products interpolating these two. We focus more in detail to the casewhere $G$ is an easy quantum group and $H=\hat{\mathbb{Z}}_2$, the dual of thecyclic group of order two. We study subgroups of $G*\hat{\mathbb{Z}}_2$ usingcategories of partitions with extra singletons. Closely related are manyexamples of non-easy bistochastic quantum groups.

New products and $\mathbb{Z}_2$-extensions of compact matrix quantum groups