This question is pertaining to finite connected vertex-transitive graphs.
I recently read "Transitive permutation groups without semiregular subgroup" by Cameron, Giudici, Jones, Kantor, Klin, Marušič, Nowitz (publisher link; MSN review), where I found the concept of "elusive groups".
A permutation group $G$ acting on a set $X$ is called elusive if $G$ is transitive and contains no nontrivial semiregular subgroup (equivalently, no fixed-point-free element of prime order). I understand that transitive subgroups of elusive groups are elusive.
My question is: In light of the Polycirculant Conjecture, are the following statements is true?
Elusive groups cannot be the full automorphism group of any vertex-transitive graph.
Elusive groups cannot be any transitive subgroup of the full automorphism group of any vertex-transitive graph.
P.S. I am trying to understand how an elusive Group is a hindrance towards the Polycirculant Conjecture.
(Previously posted on MathSE.)