  If $f(p, n)$ is the number of pairwise nonisomorphic groups of order $p^n$,
and $g(p,n)$ is the number of groups of order $p^n$ whose automorphism group is
a $p$-group, then, for $n \leq 7$, we prove that the ratio $g(p,n)/f (p,n)$ is
bounded away from 1 as the prime $p$ grows to infinity. In addition, we provide
some data on the number of groups whose automorphism group is a group of prime
power order, for primes no larger than 11.
