Finite groups close to Frobenius groups

We study finite nonsoluble generalized Frobenius groups; i.e., the groups $G$ with a proper nontrivial normal subgroup $F$ such that each coset $Fx$ of prime order $p$, as an element of the quotient group $G/F$, consists only of $p$-elements. The particular example of such a group is a Frobenius group, given that $F$ is the Frobenius kernel of $G$, and also the Camina group.