  We study the group $QV$, the self-maps of the infinite $2$-edge coloured
binary tree which preserve the edge and colour relations at cofinitely many
locations. We introduce related groups $QF$, $QT$, $\tilde{Q}T$, and
$\tilde{Q}V$, prove that $QF$, $\tilde{Q}T$, and $\tilde{Q}V$ are of type
$\mathrm{F}_\infty$, and calculate finite presentations for them. We calculate
the normal subgroup structure and rational homology of all $5$ groups, the
Bieri--Neumann--Strebel--Renz invariants of $QF$, and discuss the relationship
of all $5$ groups with other generalisations of Thompson's groups.
