3-step group for a prime

Definition
Suppose $$G$$ is a finite group and $$p$$ is a prime number. We say that $$G$$ is a 3-step group with respect to $$p$$:


 * 1) $$O_{p,p'}(G)$$ is a Frobenius group with Frobenius kernel $$O_p(G)$$ and cyclic complement of odd order. In particular, this means that either $$p = 2$$ or $$G$$ has odd order.
 * 2) $$G = O_{p,p',p}(G)$$ and strictly contains $$O_{p,p'}(G)$$.
 * 3) $$G/O_p(G)$$ is a Frobenius group with Frobenius kernel $$O_{p,p'}(G)/O_p(G)$$.

Examples
Symmetric group:S4 is an example of a 3-step group with $$p = 2$$. If $$G = S_4$$, then:


 * $$O_p(G)$$ is the normal Klein four-subgroup of symmetric group:S4, isomorphic to the Klein four-group.
 * $$O_{p,p'}(G)$$ is A4 in S4, isomorphic to alternating group:A4. This is a Frobenius group with $$O_p(G)$$ as its Frobenius kernel and complement cyclic group:Z3.
 * $$G/O_p(G)$$ is isomorphic to symmetric group:S3, and has Frobenius kernel A3 in S3, which is the image mod $$O_p(G)$$ of A4 in S4, which is $$O_{p,p'}(G)$$.