Intersection density of transitive groups with small cyclic point stabilizers

For a permutation group $G$ acting on a set $V$, a subset $F$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in F$ there exists $v\in V$ such that $g\left(v\right)=h\left(v\right)$. The intersection density $\rho \left(G\right)$ of a transitive permutation group $G$ is the maximum value of the quotient $\left|F\right|/\left|G_v\right|$ where $G_v$ is a stabilizer of a point $v\in V$ and $F$ runs over all intersecting sets in $G$. If $G_v$ is a largest intersecting set in $G$ then $G$ is said to have the Erdős-Ko-Rado (EKR)-property. This paper is devoted to the study of transitive permutation groups, with point stabilizers of prime order with a special emphasis given to orders 2 and 3, which do not have the EKR-property. Among others, constructions of an infinite family of transitive permutation groups having point stabilizer of order 3 with intersection density $4/3$ and of infinite families of transitive permutation groups having point stabilizer of order 3 with arbitrarily large intersection density are given.