Using a modification of the Kramer-Mesner method, 139 designs with pairwise distinct parameters and with a projective symplectic group as an automorphism group (i.e., as a subgroup of the full automorphism group) are constructed. Among them, there are 101 2-designs and two 3-designs over 15 points with $PS_p(4,2)$ as an automorphism group and 36 2-designs over 40 points with $PS_p(4,3)$ as an automorphism group. In particular, each of the two groups gives a Steiner system for $t=2$. Multiple appearances of the constructed designs in orbit incidence matrices are counted.