  A finite graph $\G$ is said to be {\em $(G,3)$-$($connected$)$ homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for
characterising $(G,3)$-connected homogeneous graphs. It is shown that for a
finite $(G,3)$-connected homogeneous graph $\G=(V, E)$, either $G_v^{\G(v)}$ is
$2$--transitive or $G_v^{\G(v)}$ is of rank $3$ and $\G$ has girth $3$, and
that the class of finite $(G,3)$-connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where $G$ is
quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$
in this case and give new constructions of examples for some possible types.
