Semidirect product and group action

Question

I want to understand the following lemma:

Let $G$ be a finite group satisfying $G = P \rtimes F$, where $P$ is a cyclic $p$-group for some prime $p$, $|F| > 1$ and $(p, |F|) = 1$. Then each element of $F$ acts on $P$ either trivially or fixed-point-freely.

In the article "An exact upper bound for sums of element orders in non-cyclic finite groups" (Marcel Herzog, Patrizia Longobardi, Mercede Maj) the authors show as follows:

This is the result mentioned above:

But I don't understand the connection between the two results.

Thanks in advance for any help.

Answer 1

Mercede Maj answered me. She said that the right reference is the Theorem 2.4 (in the section 5, Finite Groups - Daniel Gorenstein).