12.78 (1992)

(M. J. Curran).
a) Does there exist a group of order $p^6$ (here $p$ is a prime number), whose automorphism group has also order $p^6$?
b) Is it true that for $p \equiv 1 \pmod 3$, the smallest order of a $p$-group that is the automorphism group of a $p$-group is $p^7$?
c) The same question for $p = 3$ with $3^9$ replacing $p^7$.