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The computation of the cohomology for finite groups of Lie type in thedescribing characteristic is a challenging and difficult problem. In earlierwork, the authors constructed an induction functor which takes modules over thefinite group of Lie type to modules for the ambient algebraic group G. Inparticular this functor when applied to the trivial module yields a naturalG-filtration. This filtration was utilized in the earlier work to determine thefirst non-trivial cohomology class when the underlying root system is of typeA_{n} or C_{n}. In this paper the authors extend these results toward locatingthe first non-trivial cohomology classes for the remaining finite groups of Lietype (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 Coxeternumber.

On the vanishing ranges for the cohomology of finite groups of Lie type II