On a family of non-coset 2-valued groups

Within an important class of discrete-time 2-valued dynamical systems on $\mathbb{C}$ defined by polynomials, we extract a family of systems induced by the action of a 2-valued group. We study this 2-valued group and its analogues obtained by replacing $\mathbb{C}$ with an arbitrary field $F$. We establish that all of them are double coset groups. For some of them we prove non-cosetness. In the case $F=\mathbb{C}$, this yields the first example of a topologically non-coset, non-discrete topological 2-valued group.