Heineken-Liebeck theorem

Statement
Suppose $$G$$ is a finite group, and $$p$$ is a prime number. Then, there exists a finite $$p$$-group $$P$$ if nilpotence class two and exponent $$p^2$$ such that the image of the natural map:

$$\operatorname{Aut}(P) \to \operatorname{Aut}(P/Z(P))$$

is isomorphic to $$G$$.

Proof idea
The proof has two main parts:


 * 1) Given a finite group $$G$$, construct a strongly connected finite directed graph $$D$$ such that $$G$$ is isomorphic to the automorphism group of $$D$$.
 * 2) Given a strongly connected directed graph $$D$$ and a prime number $$p$$, construct a $$p$$-group $$P$$ of class two and exponent $$p^2$$ such that the image of $$\operatorname{Aut}(P)$$ in $$\operatorname{Aut}(P/Z(P))$$ has exponent dividing $$p^2$$.

Textbook references

 * , Page 404, Section 13.6