  We define a family of groups that generalises Thompson's groups $T$ and $G$
and also those of Higman, Stein and Brin. For groups in this family we descrine
centralisers of finite subgroups and show, that for a given finite subgroup
$Q$, there are finitely many conjugacy classes of finite subgroups isomorphic
to $Q$. We use this to show that whenever defined, the T versions of these
groups have a slightly weaker property, quasi-$\underline{\operatorname
F}_\infty$, to that of a group possessing a finite type model for the
classifying space for proper actions ${\underline{E}}G$ if and only if they
posses finite type models for the ordinary classifying space. We also
generalise some well-known properties of ordinary cohomology to Bredon
cohomology.
