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

There are two very natural products of compact matrix quantum groups: the tensor product $G\times H$ and the free product $G\ast H$. We define a number of further products interpolating these two. We focus more in detail to the case where $G$ is an easy quantum group and $H={\widehat{\mathbb{Z}}}_2$, the dual of the cyclic group of order two. We study subgroups of $G\ast {\widehat{\mathbb{Z}}}_2$ using categories of partitions with extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.