Linear $2$-Group Actions With Exactly One Regular Orbit

Let $G$ be a finite $2$-group and $V$ a finite faithful irreducible $G$-module. We show that if $G$ has exactly one regular orbit on $V$, then the characteristic $q$ of $V$ is a Fermat prime, $V$ is irreducible over ${\mathbb{F}}_q$, ${\mathrm{d}\mathrm{i}\mathrm{m}}_{{\mathbb{F}}_q}V$ is a power of $2$, and the centraliser of $G$ in $\mathrm{G}\mathrm{L}(V)$ embeds into $G$. For every Fermat prime $q$ and every $m\ge 1$ we determine the largest orbit size $2^{x_m}$ of a Sylow $2$-subgroup $P_m$ of ${\mathrm{G}\mathrm{L}}_{2^{m-1}}(q)$ on its natural module $V_m$, and the number of orbits of this size, which is $1$, $2$, or $3$. Our main result is that for every Fermat prime $q$ and every $m\ge 1$ there is an irreducible subgroup of $P_m$ of order $2^{x_m}$ that has exactly one regular orbit on $V_m$. In this respect $p=2$ behaves like $p=3$, and unlike the Mersenne primes $p\ge 7$ studied in an earlier paper. The proof places a group related to a Singer cycle into every block of $q-1$ coordinates. This reduces the problem, for all Fermat primes at once, to a question about subgroups of a Sylow $2$-subgroup of a symmetric group acting on subsets, which we answer by means of fibre products. We also determine which largest orbits occur in two small levels, and we classify the examples in some small cases with the help of a computer.