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Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this?

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.

Easy example of a $2$-Frobenius group is $S_4$, where a $2$-cycle acts fixed point freely on the $3$-cycles, and the $3$ cycles act fixed point freely on the $4$-cycles. In general it's easy to make $2$-Frobenius groups like this by selecting cyclic groups of appropriate orders, but there are other examples too. One can interpret also this by taking a faithful representation of a Frobenius group over a finite field with $0$-dimensional fixed point space from the kernel.

Question 1:

I've seen some theorems about characters of Frobenius groups in Huppert's and Isaccs' character theory books, but nothing specifically about $2$-Frobenius groups. In fact there is a specific formula for the characters of a Frobenius group $G=FH$ with kernel $F$:

The characters of $G=FH$ are

  1. The characters in $\text{Irr}\,H$ are characters of $G$ with $F$ in their kernel.
  2. Define an action of $H$ on $\text{Irr}\,F$ by $\phi^h(f)=\phi(f^{h^{-1}})$. The orbits of $H$ on $\text{Irr}\,F$ are $\{\phi_1=1_F\},\{\phi^h_j|h\in H\}$ for $j=2,\ldots,(1+|\text{Irr}\,F|-1)/|H|$. Then the $\phi_j^G$ are irreducible on $G$.

(Thus, a corollary: $|\text{Irr}\,G|=|\text{Irr}\,H|+\frac{|\text{Irr}\,F|-1}{|H|}$.)

Could this be adapted to describe the characters of $2$-Frobenius groups? I'm particularly interested in Frobenius groups where $F_1$ and $G/F_2$ are $p$-groups and $F_2/F_1$ is a $q$-group (for distinct $p$,$q$).

Question 2:

Is there any information out there in the literature about characters of $2$-Frobenius groups? (Maybe some papers?)

If not, can anyone think of other Frobenius group character theorems that could be adapted smoothly to $2$-Frobenius groups? Doesn't matter how tedious or specific; anything to help me understand this type of group better.

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