On designs constructed by group actions

Using the action of the linear fractional groups \( L_2(q) \), where \( q = 8, 25, 27, 29, 31 \) and \( 32 \), some \( 1 \)-designs are constructed. It is shown that subgroups of the automorphism group of \( L_2(q) \) appear as the full automorphism group of the constructed designs. In the cases \( q = 8 \) and \( 32 \), it is shown that the symmetric groups \( \mathbb{S}_9 \) and \( \mathbb{S}_{33} \), respectively, appear as the automorphism group of one of the constructed designs.