2-Frobenius groups of order $2^{10}.3^5.5.11$ A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with kernel $B$ and complement $C$. 
Let  $A$ be a nilpotent group of order $2^{10}.3^5$ and order $B$  equal to $11$ and order $C$ equal to $5$. So we know that $BC$ is the only Frobenius group of order $55$. Now, I would like to know that whether we can find all such 2-Frobenius group $G=ABC$ of order $2^{10}.3^5.5.11$?
 A: I don't want to write a detailed solution but here are a few guidelines.
Any such group will be a subdirect product of $2$-Frobenius groups of orders $2^{10}.11.5$ and $3^5.11.5$, so you might as well deal with those two orders eparately.
Since $10$ is the order of $2$ modulo $11$, $10$ is the smallest dimension of an irreducible module for $C_{11}$ over ${\mathbb F}_2$, so there is a unique Frobeius group of order $2^{10}.11$ and its kernel is elementary abelian. This is a subgroup of the group ${\rm AGL}(1,2^{10})$. You can extend this by an element of order $5$ acting as a field automorphism of ${\mathbb F}_{2^{10}}$, giving a unique isomorphism class of $2$-Frobenius groups of order $2^{10}.11.5$. Another way of looking at it is that the normalizer of a
subgroup of order $11$ in ${\rm GL}(10,2)$ is the semilinear group, which is metacyclic with structure $C_{1023}:C_{10}$, and this has the Frobenius group $C_{11}:C_5$ as subgroup.
By similar reasoning you can show that there is a unique $2$-Frobenius group of order $3^5.11.5$, and it has elementary abelian normal subgroup of order $3^5$. In this case there are two isomorphism types of modules of dimension $5$ over ${\mathbb F}_3$, but only one isomorphism type of groups, 
Taking the subdirect product of these gives a unique isomorphism class of $2$-Frobenius groups of order $2^{10}.3^5.11.5$.
(You have to be cautious in this situation. Unique isomorphism classes of the two subdirect factors does not necessarily imply a unique isomorphism type of subdirect products of the required structure, but it does in this case.)
