The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich:
Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one. ($p$ as usual denotes a prime number)
I would like to know if anybody knows anything concerning history of this problem.
What I know are as follows: W. Gaschütz has proved in 1966 that every finite $p$-group of order greater than $p$ has a non-inner automorphism of $p$-power order. One year before Gaschütz, H. Liebeck has proved problem 4.13 for $p$-groups of class $2$ whenever $p>2$. The problem in its own is of course interesting, but I think maybe the main motivation to propose the problem is to strengthen Gaschütz's famous result, am I right?
What I really look for is to know: does the problem come from other mathematics problems that people already noted to and they stated it? For example, has it any relation to the classification of finite simple groups?!
I am sorry if you feel my questions are so vague! and I apologize in advance for.