Let $G=KH$ be a frobenius group with non-abelian kernel $K$,

$|H|=r-1$,

$|K|=r^2$,

$r=2^m$ for some odd integer $m$,

$Z(K)=K'=\Phi(K)$, the Frattini subgroup of $K$,

$[K:K']=|K'|=r$.

Let both $K/K'$ and $K'$ be elementary abelian 2-groups.

My questions are:

1. In my special case, is it correct that {$k\in K| k^2=1$}$=:\Omega_1(K)\neq K$ ?

2. I don't know much about $\Omega_1(K)$. Concerning my first question, are there any helpful theorems?

Let $G=KH$ be a frobenius group with non-abelian kernel $K$,

$|H|=r-1$,

$|K|=r^2$,

$r=2^m$ for some odd integer $m$,

$Z(K)=K'=\Phi(K)$, the Frattini subgroup of $K$,

$[K:K']=|K'|=r$.

Let both $K/K'$ and $K'$ be elementary abelian 2-groups.

My questions are:

1. In my special case, is it correct that <{$k\in K| k^2=1$}$>=:\Omega_1(K)\neq K$ ?

2. I don't know much about $\Omega_1(K)$. Concerning my first question, are there any helpful theorems?