Every finite group is the Fitting quotient of a p-dominated group for any prime p not dividing its order

History
This result was proved in a paper by Hartley and Robinson.

Statement
Suppose $$H$$ is a finite group and $$p$$ is a prime not dividing the order of $$H$$. Then, there exists a fact about::p-dominated group whose quotient by its Fitting subgroup is $$H$$.

In other words, there is a fact about::finite complete group $$G$$ (i.e., a finite group that is complete: it is centerless and every automorphism is inner) such that the fact about::Fitting subgroup $$F(G)$$ is a group of prime power order for the prime $$p$$, and the quotient group $$G/F(G)$$ is isomorphic to $$H$$. In fact, $$G$$ is the semidirect product of $$F(G)$$ and $$H$$.

Facts used

 * 1) uses::Bryant-Kovacs theorem