$G$ finite group of order $2^kp$ and $G_p:=\{g\in G\mid g^p=e\}$: Is it $\#G_p\le p^2-1$?

Question

Let $G$ be a finite group of order $2^kp$, for some odd prime $p$ and integer $k>1$, and let's define $G_p:=\{g\in G\mid g^p=e\}$.

Is it $\space\#G_p\le p^2-1$?

I'm aware that in the case $G=\operatorname{GL}_2(\mathbb{F}_p)$ the "$=$" sign holds, but at the cost of making use of quite advanced stuff (e.g. Sylow theory, Jordan Canonical Form Theorem). I'd be fine with the weaker result in the question, if this would come with more elementary tools.

Answer 1

No, for example there is a Frobenius group of order 48 with 32 elements of order 3.

The correct bound is $\#G_p \le 2^k(p-1)$.