  The computation of the cohomology for finite groups of Lie type in the
describing characteristic is a challenging and difficult problem. In earlier
work, the authors constructed an induction functor which takes modules over the
finite group of Lie type to modules for the ambient algebraic group G. In
particular this functor when applied to the trivial module yields a natural
G-filtration. This filtration was utilized in the earlier work to determine the
first non-trivial cohomology class when the underlying root system is of type
A_{n} or C_{n}. In this paper the authors extend these results toward locating
the first non-trivial cohomology classes for the remaining finite groups of Lie
type (i.e., the underlying root system is of type B_{n}, C_{n}, D_{n}, E_{6},
E_{7}, E_{8}, F_{4}, and G_{2}) when the prime is larger than the Coxeter
number.
