  A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is
adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to
be $2$-geodesic transitive if its automorphism group is transitive on the set
of $2$-geodesics. In this paper, a complete classification of $2$-geodesic
transitive graphs of order $p^n$ is given for each prime $p$ and $n\leq 3$. It
turns out that all such graphs consist of three small graphs: the complete
bipartite graph $K_{4,4}$ of order $8$, the Schl\"{a}fli graph of order $27$
and its complement, and fourteen infinite families: the cycles $C_p, C_{p^2}$
and $C_{p^3}$, the complete graphs $K_p, K_{p^2}$ and $K_{p^3}$, the complete
multipartite graphs $K_{p[p]}$, $K_{p[p^2]}$ and $K_{p^2[p]}$, the Hamming
graph $H(2,p)$ and its complement, the Hamming graph $H(3,p)$, and two infinite
families of normal Cayley graphs on extraspecial group of order $p^3$ and
exponent $p$.
