Primary cosets in groups

A finite group $G$ is called a generalized Frobenius group with kernel $F$ if $F$ is a proper nontrivial normal subgroup of $G$, and for every element $Fx$ of prime order $p$ in the quotient group $G/F$, the coset $Fx$ of $G$ consists of $p$-elements. We study generalized Frobenius groups with an insoluble kernel $F$. It is proved that $F$ has a unique non-Abelian composition factor, and that this factor is isomorphic to $L_2(3^{2^l})$ for some natural number $l$. Moreover, we look at a (not necessarily finite) group generated by a coset of some subgroup consisting solely of elements of order three. It is shown that such a group contains a nilpotent normal subgroup of index three.