A finite group $B$ is said to be a B-group if every primitive permutation group having a regular (transitive) subgroup isomorphic to B is $2$-transitive.
Schur proved that a cyclic group of composite order is a B-group. Wielandt showed that no group of the form $B_1 \times \cdots \times B_d$ with $|B_1| = \ldots = |B_d| \ge 3$ and $d \ge 2$ is a B-group. (See Theorems 25.3 and 25.7 in Wielandt, Finite permutation groups.)
Exercise 3.5.6 in Dixon and Mortimer Permutation groups asks for a proof that no elementary abelian $p$-group is a B-group. However this is false: the permutation groups of degree $4$ containing $C_2 \times C_2$ as a regular subgroup are $\langle (12)(34), (13)(24)\rangle$ and $\mathrm{Dih}(4)$ (both imprimitive) and $A_4$, $S_4$ (both $2$-transitive). My question is:
For which $d \in \mathbb{N}$ is the elementary abelian group $C_2^d$ a B-group?
Necessary conditions for a primitive permutation groups to have a regular subgroup are given in Corollary 3 of Liebeck, Praeger, Saxl, Transitive Subgroups of Primitive Permutation Groups, J. Alg 234, 291–361 (2000). The main theorem of Li, The finite primitive permutation groups containing an abelian regular subgroup, Proc. London Math. Soc. 87, 725–747 (2003), gives a sharper result. It seems non-trivial to use either of these papers to answer the question, but I'd welcome a correction.
If $H \le \mathrm{GL}(\mathbb{F}_2^d)$ then the subgroup $\mathbb{F}_2^d \rtimes H$ of the affine general linear group $\mathrm{AGL}(\mathbb{F}_2^d)$ is primitive if and only if $H$ is irreducible. So if there exists an irreducible $H$ acting intransitively on $\mathbb{F}_2^d \backslash \{0\}$, then $C_2^d$ is not a B-group. Such groups exist whenever $d$ is composite, but not always when $d$ is prime. For example, the three irreducible subgroups of $\mathrm{GL}(\mathbb{F}_2^3)$ all contain a Singer element of order $7$, so give $2$-transitive subgroups of $\mathrm{AGL}(\mathbb{F}_2^3)$. (The only other primitive permutation groups of degree $8$ containing a subgroup isomorphic to $C_2 \times C_2 \times C_2$ are $A_8$ and $S_8$, so it follows that $C_2 \times C_2 \times C_2$ is a B-group.) This motivates my second question:
For which primes $p$ is there an irreducible subgroup $H$ of $\mathrm{GL}(\mathbb{F}_2^p)$ such that $H$ acts intransitively on $\mathbb{F}_2^p \backslash \{0\}$?