Finite $3$-connected-set-homogeneous locally $2\K_n$ graphs and $s$-arc-transitive graphs

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.