Summary: The class \(\mathbf H\mathfrak F\) of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever \(G\) admits a finite-dimensional contractible \(G\)-CW-complex in which all stabilizer groups are in \(\mathbf H\mathfrak F\), then \(G\) is itself in \(\mathbf H\mathfrak F\). The class \(\mathbf H\mathfrak F\) admits a natural filtration indexed by the ordinals. For example, \(\mathbf H_0\mathfrak F\) is the class of all finite groups and \(\mathbf H_1\mathfrak F\) contains all groups of finite virtual cohomological dimension. We show that, for each countable ordinal \(\alpha\), there is a countable group that is in \(\mathbf H\mathfrak F\setminus\mathbf H_\alpha\mathfrak F\). Previously this was known only for \(\alpha=0\), \(1\) and \(2\). The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in \(\mathbf H\mathfrak F\setminus\mathbf H_2\mathfrak F\).