  We construct a differential graded algebra (DGA) modelling certain $A_\infty$
algebras associated with a finite group $G$ with cyclic Sylow subgroups, namely
$H^*BG$ and $H_*\Omega BG^{^\wedge}_p$. We use our construction to investigate
the singularity and cosingularity categories of these algebras. We give a
complete classification of the indecomposables in these categories, and
describe the Auslander--Reiten quiver. The theory applies to Brauer tree
algebras in arbitrary characteristic, and we end with an example in
characteristic zero coming from the Hecke algebras of symmetric groups.
