Summary: ``All quasi-symmetric $2$-$(28,12,11)$ designs with an automorphism of order $7$ without fixed points or blocks are enumerated. Up to isomorphism, there are exactly $246$ such designs. All but four of these designs are embeddable as derived designs in symmetric $2$-$(64,28,12)$ designs, producing in this way at least $8784$ nonisomorphic symmetric $2$-($64,28,12)$ designs. The remaining four $2$-$(28,12,11)$ designs are the first known examples of nonembeddable quasi-symmetric quasi-derived designs. These symmetric $2$-$(64,28,12)$ designs also produce at least $8784$ nonisomorphic quasi-symmetric $2$-$(36,16,12)$ designs with intersection numbers $6$ and $8$, including the first known examples of quasi-symmetric $2$-$(36,16,12)$ designs with a trivial automorphism group.''