Flag-transitive 2-designs from PSL(2, q) with block size 4

This paper is a contribution to the classification of flag-transitive 2-(v,4,λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v,4,\lambda )$$\end{document} designs. We determine all such designs admitting G=PSL(2,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=PSL(2,q)$$\end{document} as a flag-transitive automorphism group.