The author constructs a symmetric (78,22,6)-design with a block B such that the intersection of the blocks \(\neq B\) induce on B the Witt design S(3,6,22) thus solving a problem of D. R. Hughes [”The non- existence of a semi-symmetric 3-design with 78 points”, Ann. Discrete Math. 18, 473-479 (1983; Zbl 0521.05014)]. The full automorphism group of the author’s design is of order 168 (and is a maximal subgroup of \(2^ 3\cdot PSL(2,3)\) which is contained in \(M_{22}\) Aut S(3,6,22). The author also shows that S(3,6,22) cannot be embedded in a symmetric (78,22,6)-design in such a way that \(M_{22}\) or one of its maximal subgroups \(M_{21}\), \(2^ 4A_ 6\), PSL(2,11) or \(2^ 3PSL(3,2)\) are automorphism groups of the embedding. Only one example of a symmetric (78,22,6)-design was known previously, see Z. Janko and Tran van Trung: ”Construction of a new symmetric block design for (78,22,6) with the help of tactical decompositions” [J. Comb. Theory, Ser. A 40, 451-455 (1985; Zbl 0577.05011)]; this design does not carry an induced S(3,6,22). Since Tonchev’s design is not self-dual, we now have at least 3 non-isomorphic symmetric (78,22,6)-designs.