For a non-complete graph $Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u eq w$ and $u,w$ are not adjacent. Then $Gamma$ is said to be $2$-geodesic transitive if its automorphism group is transitive on both arcs and 2-geodesics. In previous work the author showed that if a $2$-geodesic transitive graph $Gamma$ is locally disconnected and its automorphism group $Aut(Gamma)$ has a non-trivial normal subgroup which is intransitive on the vertex set of $Gamma$, then $Gamma$ is a cover of a smaller 2-geodesic transitive graph. Thus the `basic graphs to study are those for which $Aut(Gamma)$ acts quasiprimitively on the vertex set. In this paper, we study 2-geodesic transitive graphs which are locally disconnected and $Aut(Gamma)$ acts quasiprimitively on the vertex set. We first determine all the possible quasiprimitive action types and give examples for them, and then classify the family of $2$-geodesic transitive graphs whose automorphism group is primitive on its vertex set of $PA$ type.
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