  In this paper, all graphs are assumed to be finite. For $s\geq 1$ and a graph
$\G$, if for every pair of isomorphic connected induced subgraphs on at most
$s$ vertices there exists an automorphism of $\G$ mapping the first to the
second, then we say that $\G$ is $s$-connected-set-homogeneous, and if every
isomorphism between two isomorphic connected induced subgraphs on at most $s$
vertices can be extended to an automorphism of $\G$, then we say that $\G$ is
$s$-connected-homogeneous. For $n\geq 1$, a graph $\G$ is said to be locally
$2\K_n$ if the subgraph $[\G(u)]$ induced on the set of vertices of $\G$
adjacent to a given vertex $u$ is isomorphic to $2\K_n$.
  Note that $2$-connected-set-homogeneous but not $2$-connected-homogeneous
graphs are just the half-arc-transitive graphs which are a quite active topic
in algebraic graph theory. Motivated by this, we posed the problem of
characterizing or classifying $3$-connected-set-homogeneous graphs of girth $3$
which are not $3$-connected-homogeneous in (Eur. J. Combin. 93 (2021) 103275).
Until now, there have been only two known families of
$3$-connected-set-homogeneous graphs of girth $3$ which are not
$3$-connected-homogeneous, and these graphs are locally $2\K_n$ with $n=2$ or
$4$. In this paper, we complete the classification of finite
$3$-connected-set-homogeneous graphs which are locally $2\K_n$ with $n\geq 2$,
and all such graphs are line graphs of some specific $2$-arc-transitive graphs.
Furthermore, we give a good description of finite $3$-connected-set-homogeneous
but not $3$-connected-homogeneous graphs which are locally $2\K_n$ and have
solvable automorphism groups. This is then used to construct some new
$3$-connected-set-homogeneous but not $3$-connected-homogeneous graphs as well
as some new $2$-arc-transitive graphs.
