Flag-transitive 4-designs and PSL(2, q) groups

This paper is a contribution to the classification of flag-transitive 4- $$(v,k,\lambda )$$ designs. Let $$\mathcal D=({\mathcal {P}}, {\mathcal {B}})$$ be $$(q+1,k,\lambda design with $$\lambda \ge 5$$ and $$q+1>k>4$$ , $$G=PSL(2,q)$$ automorphism group D$$ $$G_x$$ stabilizer point $$x\in {P}}$$ $$G_B$$ setwise block $$B\in {B}}$$ . Using fact that must one twelve kinds subgroups PSL(2, q), up isomorphism we get following two results: (i) If $$10\ge \lambda then possible exception $$(G,G_x,G_B,k,\lambda )=(PSL(2,761),{E_{761}}\rtimes {C_{380}},S_4,24,7)$$ or $$(PSL(2,512),{E_{512}}\rtimes {C_{511}},{D_{18}},18,8)$$ which remain undecided, unique 4-(24, 8, 5), 4-(9, 6, 10), 4-(8, 6), 7, 4-(10, 9, 4-(12, 11, 8) 4-(14, 13, 10) $$(G,G_x,G_B)=(PSL(2,23),$$ $${E_{23}}\rtimes {C_{11}},D_8)$$ $$(PSL(2,8),{E_{8}}\rtimes {C_{7}},D_6)$$ $$(PSL(2,7),{E_{7}}\rtimes {C_{3}},D_6)$$ {C_{7}},D_{14})$$ {C_{7}},{E_8}\rtimes {C_7})$$ $$(PSL(2,9),{E_{9}}\rtimes {C_{4}},{E_9}\rtimes {C_4})$$ $$(PSL(2,11),{E_{11}}\rtimes {C_{5}},{E_{11}}\rtimes {C_{5}})$$ $$(PSL(2,13),{E_{13}}\rtimes {C_{6}},{E_{13}}\rtimes {C_6})$$ respectively. (ii) >10$$ $${G_B}=A_4$$ $$S_4$$ $$A_5$$ $$PGL(2,q_0)$$ ( $$g>1$$ even) $$PSL(2,q_0)$$ where $${q_0}^g=q$$ there no such