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

fullautomorphism group of any vertex-transitive graph.

or

Elusive groups cannot be any transitive subgroup of the

fullautomorphism 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.)