Feit & Thompson (1963, Chapter 3) developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable.
Feit proved several theorems giving conditions under which a set of characters is coherent.
If G is the simple Suzuki group of order (2n–1) 22n( 22n+1) with n odd and n>1 and H is the Sylow 2-subgroup and τ is induction, then coherence fails for the second reason.
In the proof of the Frobenius theory about the existence of a kernel of a Frobenius group G where the subgroup H is the subgroup fixing a point and S is the set of all irreducible characters of H, the isometry τ on I0(S) is just induction, although its extension to I(S) is not induction.
Similarly in the theory of exceptional characters the isometry τ is again induction.