Many common finite p-groups admit non-trivial automorphisms of order coprime to p,
However when p is odd, it is reasonably difficult to find finite p-groups with
automorphism group a p-group.  I'll talk about joint work with Ursula
Martin proving that almost all finite p-groups do have automorphism group a
p-group when p is odd.  The asymptotic sense in which the theorem holds
involves bounding the Frattini length of the p-groups and letting the
number of generators go to infinity.  The proof of the theorem draws on a
detailed analysis of the Frattini series of a free group and the
combinatorics linking finite p-groups and representations of GL(n,p). 
The case of p = 2 remains open.