  Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is
$2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a
derangement of odd prime order. In this paper we study the structure of
quasiprimitive and biquasiprimitive $2'$-elusive permutation groups, extending
earlier work of Giudici and Xu on elusive groups. As an application, we use our
results to investigate automorphisms of finite arc-transitive graphs of prime
valency.
