It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley-Hadamard matrix of type I. If $q=5, 11, 17, 23$, then the full permutation automorphism group of $L(q)$ coincides with the full automorphism group of $D(q)$, and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code $C(17)$ are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley-Hadamard matrix $H$ of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix $H'$ such that the symmetric 2-$(36,15,6)$ design $D$ associated with $H'$ has trivial full automorphism group, and the incidence matrix of $D$ spans a ternary code equivalent to $C(17)$.