Order of elements from coset and Schur-Zassenhaus Theorem (Exercise $3.B.6$ FGT)

Question

I try to solve exercise 3.B.6 from M. Isaacs's book Finite Group Theory (FGT). I give it here.

Let $N \vartriangleleft G$ and $g\in G$, and suppose that when $Ng$ is viewed as an element of $G/N$, it has order $m$

1. Show that there exists $h\in Ng$ such that every prime divisor of $o(h)$ divides m. Here $o(h)$ is the order of $h$.
2. If $m$ and $|N|$ are coprime, show that the element $h$ of the first part must have order $m$
3. Now assume that $Ng$ is conjugate to its inverse in $G/N$ and that $m$ and $|N|$ are coprime. Let $h\in Ng$ be as in the first part. Show that $h$ is conjugate to its inverse in $G$.

There is a hint to the first and the third part of this problem. I give it here too.

For the first part find a cyclic $\pi$-subgroup $C$ such that
$NC = N\langle g \rangle$, where $\pi$ is the set of prime divisors of $m$. For the third part, let $x\in G$, where $(Ng)^{Nx} = Ng^{-1}$. Observe that $x$ normalizes $N\langle h \rangle$. Consider the subgroup $\langle h \rangle^x$.

I can do only the first part of this problem. However I don't understand how to use given hint for the first part. Also I stuck with the second and the third part. I suggest that we should apply the Schur-Zassenhaus Theorem here. I will be grateful for ideas and hints.