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 $Gtimes H$ and the free product $G*H$. 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 $H=hat{mathbb{Z}}_2$, the dual of the cyclic group of order two. We study subgroups of $G*hat{mathbb{Z}}_2$ using categories of partitions with extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.