Elusive groups from non-split extensions

A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts that no elusive group is $2$-closed. Existing constructions of elusive groups mostly arise from split extensions. In this paper, we initiate the construction of elusive groups via non-split extensions. This approach offers a key advantage: it allows the construction of elusive groups even when the acting group is not elusive. As a demonstration, we construct elusive groups of new degrees, namely p³ᵏ⁻⁴(p+1)/2 for each Mersenne prime p≥7 and integer k≥2, along with the first odd degree $225$ and the first twice odd degree $450$. We conclude by proposing further problems to advance this new direction of research.