A search for those$t-(q+1,k,\lambda)$ designs is made, which arise by action of the groups $PSL(2,q)$ and $PGL(2,q)$ on the ground-set $\Omega(q)=\{0,1,\ldots,q-1\}\cup\{\infty\}$. The search is made for $(t,k)=(4,5)$ with prime powers $q\leq 49$ and for $(t,k)\in\{(4,6),(5, 6)\}$ with prime powers $q\leq31$. The group $PSL(2,q)$ is used for $q\equiv3$ (mod 4) and the group $PGL(2,q)$ is used otherwise. The search uses orbit incidence matrices determined by orbits of $t$-subsets and $t$-subsets (shortly: $t$-orbits and $k$-orbits) of the ground-set, obtained by action of the group used. An element of an orbit incidence matrix is the number of those $k$-sets within a $k$-orbit, which contain a fixed $t$-set (representative) of a $t$-orbit. Construction of orbit incidence matrices essentially uses 3-homogenicity of the groups. The total number of distinct quadruples $(t,q,k,\lambda)$ of parameters, for which $t-(q+q,k,\lambda)$ designs are constructed is equal to 75. It is guaranteed that the obtained values of $\lambda$ are the only possible, which can be reached by action of the groups used, for the considered triples $(t,q,k)$. It Is assumed that most of the obtained quadruples of design parameters are new, in particular those for $q=19,25,27,31$ and 37.