  We give a construction of a family of designs with a specified
point-partition, and determine the subgroup of automorphisms leaving invariant
the point-partition. We give necessary and sufficient conditions for a design
in the family to possess a flag-transitive group of automorphisms preserving
the specified point-partition. We give examples of flag-transitive designs in
the family, including a new symmetric $2$-$(1408,336,80)$ design with
automorphism group $2^{12}:((3\cdot\mathrm{M}_{22}):2)$, and a construction of
one of the families of the symplectic designs (the designs $S^-(n)$) exhibiting
a flag-transitive, point-imprimitive automorphism group.
