Vertex-imprimitive symmetric graphs with exactly one edge between any two distinct blocks

A graph $\mathrm{\Gamma }$ is called $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of $G$-symmetric graphs $\mathrm{\Gamma }$ with $V(\mathrm{\Gamma })$ admitting a nontrivial $G$-invariant partition $\mathcal{B}$ such that there is exactly one edge of $\mathrm{\Gamma }$ between any two distinct blocks of $\mathcal{B}$. This is achieved by giving a classification of $(G, 2)$-point-transitiveand$G$-block-transitivedesigns$$\mathcal{D}$$ together with $G$-orbits $\mathrm{\Omega }$ on the flag set of $\mathcal{D}$ such that $G_{\sigma, L}$istransitiveon$L \setminus {\sigma}$and$L \cap N = {\sigma}$ for distinct $(\sigma ,L),(\sigma ,N)\in \mathrm{\Omega }$, where $G_{\sigma ,L}$ is the setwise stabilizer of $L$ in the stabilizer $G_{\sigma }$ of $\sigma$ in $G$. Along the way we determine all imprimitive blocks of $G_{\sigma }$ on $V \setminus {\sigma}$forevery$2$-transitivegroup$G$onaset$V$,where$\sigma \in V$.