  We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an
Out(P)--module. In particular, we study the conjecture, first considered by
Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent
associated to an irreducible Z/p[Out(P)]--module S, then the Krull dimension of
e_SH^*(P) equals the rank of P. The rank is an upper bound by Quillen's work,
and the conjecture can be viewed as the statement that every irreducible
Z/p[Out(P)]--module occurs as a composition factor in H^*(P) with similar
frequency.
  In summary, our results are as follows. A strong form of the conjecture is
true when p is odd. The situation is much more complex when p=2, but is reduced
to a question about 2--central groups (groups in which all elements of order 2
are central), making it easy to verify the conjecture for many finite
2--groups, including all groups of order 128, and all groups that can be
written as the product of groups of order 64 or less.
  Featured is the nilpotent filtration of the category of unstable A--modules.
Also featured are unstable algebras of cohomology primitives associated to
central group extensions.
