Definition: We say that $g\in G$ is a vanishing element if there exists $\chi \in \text{Irr(G)}$ such that $\chi(g)=0$.
Lemma 1: Let $N \leq M \leq G$ with $N$ and $M$ normal in $G$ and $(|N|, |M/N| ) =1$. If $N$ is a minimal normal subgroup of $G$, $C_M(N) \leq N$ and $M/N$ is abelian, then every element in $M\backslash N$ is vanishing in $G$.
Lemma 2: Let $G$ be a group with abelian minimal normal subgroup $K$. Let $M/N$ be a chief factor of $G$ such that $(|K|, |M/N| ) = 1$ and $N =C_M(K)$. Then every element in $M\backslash N$ is vanishing in $G$.
I am having a problem understanding a part of proof of a result from a paper. I want to make use of the lemmas above.
Let $G$ be a finite solvable group with minimal normal subgroup $N$. Suppose that $K$ and $H$ are subgroups such that $K/N$ is a chief factor of $G$ and $K \leq NH$. Furthermore, we may assume that $H$ acts faithfully on $N$ and that $H\cap N ={1}$. If $\psi \in \text{Irr}(K)$, then $\psi^g$ vanishes on every $1\neq x \in K \cap H$ for all $g\in G$
Here's what I know so far:
Since $G$ is solvable, we have $K/N$ is an abelian $p$-group for some prime $p$. $(K\cap H)/\{1\} \cong K \cap H $ is isomorphic to a subgroup of $K/N$. Hence, $K\cap H$ is an abelian $p$-group. It is clear that $N$ is a $p'$-group, so that $(|N|, |K/N| ) =1$. Since $H$ acts faithfully on $N$, we have that $C_H(N) = 1$. Then $C_{K\cap H}(N) = C_H(N) \cap K\cap H = 1$. Thus, $K\cap H$ acts faithfully on $N$. If I try to use Lemma 1 to get a deduction, I'm uncertain that $C_K(N) \leq N$. If I try to use Lemma 2 with $M = K \cap H$, $G=K$, then I'm unable to show that $ (K\cap H)/\{1\}$ is a chief factor of $K$