In this paper we analyze possible actions of an automorphism of order six on a $2$-$(70, 24, 8)$ design, and give a complete classification for the action of the cyclic automorphism group of order six $G= \langle ρ\rangle \cong Z_6 \cong Z_2 \times Z_3$ where $ρ^3$ fixes exactly $14$ points (blocks) and $ρ^2$ fixes $4$ points (blocks). Up to isomorphism, there are $3718$ such designs. This result significantly increases the number of known $2$-$(70,24,8)$ designs.