A $3$-$(v,\{4,6\},1)$ design is a configuration of $v$ points and a collection of $4$- and $6$-element subsets called blocks, that jointly contain every 3-element subset exactly once. Using an exhaustive computer search on $v\leq 28$ points we investigate the $3$-$(v,\{4,6\},1)$ designs that have a transitive automorphism group and where the blocks of size 6 form a 2-class symmetric design. A 2-class symmetric design with parameters $(v,k{;}λ_1,λ_2{;}δ_1,δ_2)$ is a set-system on $v$ points and $v$ blocks of size $k$, where every pair of points are in $λ_1$ or $λ_2$ blocks and every pair of blocks intersect in $δ_1$ or $δ_2$ points. The 2-class symmetric designs include biplanes, semi-biplanes, and 2-class symmetric partially balanced incomplete block designs.