This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.

Definition.Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/F_1\right)$. If $F_2$ is a Frobenius group with kernel $F_1$ and $G/F_1$ is a Frobenius group with kernel $F_2/F_1$, we say that $G$ is$2$-Frobenius.

I have read about the characters of Frobenius groups in Isaacs and Huppert's books, but I have never seen $2$-Frobenius groups mentioned. Can anyone point me to some literature on the character theory of $2$-Frobenius groups?

Alternatively, does anyone know any theorems about Frobenius groups that could be adapted to $2$-Frobenius groups? I am especially interested in $2$-Frobenius groups where $F_1$ and $G/F_2$ are $p$-groups and $F_2/F_1$ is a $q$-group (for distinct primes $p$,$q$), but I would appreciate any representation theory at all which may help me better understand this class of groups.