Hall's theorem and generalized Sylow theorem.

Question

Hall's theorem states that:

Let $G$ be a finite group. The following statements are equivalent:

1. $G$ is solvable.
2. $G$ is $\pi$-separable for every set of primes $\pi$.
3. $G$ contains a $\pi$-Hall subgroup for every set of primes $\pi$.
4. $G$ contains a $p'$-Hall subgroup for every primes $p$.
5. $G$ possesses a Sylow System.

I am trying to understand why the generalized Sylow theorem is a Corollary of Hall's Theorem. What I am saying is that I want to prove, using Hall's Theorem,o that:

Let $G$ be a finite solvable group, if $|G|=a \cdot b$ with $mcd(a,b)=1$ then:

1. $G$ has a subgroup $H$ such that $|H|=a$
2. Two subgroup $H_{1}$ and $H_{2}$ such that $|H_{1}|=|H_{2}|=a$ are conjugate.
3. If $H'$ is a subgroup such that $|H'|$ divides $a$ then $H'$ is contained in a subgroup of order $a$.

I was able to prove $(1)$ but I am having difficulties in proving $(2)$ and $(3)$. It is also worth saying that I know a direct proof by induction of the theorem but what I am looking for is a way to prove it using Hall's Theorem. Any help?