On primitivity and reduction for half-flag-transitive block designs

Let $\mathcal{D}=(\mathcal{P},\mathcal{B})$ be a $2$-$(v,k,\lambda )$ design, and let $G$ be a half-flag-transitive automorphism group of $\mathcal{D}$. In this article, we first establish three sufficient conditions for $G$ to be point-primitive: (i) $\lambda \ge (r,2\lambda {)}^2$, (ii) $r>4\lambda (k-2)$, (iii) $(v-1,2k-2)\le 2$. Next, we prove that for $\lambda \ge (r,2\lambda {)}^2$, the group $G$ is either of affine type, almost simple type, or product type. Finally, we analyze the case where $G$ is of almost simple type and prove that if the socle of $G$ is a sporadic simple group then $G\cong \text{HS}$ and $\mathcal{D}$ is either the unique $2$-$(176,128,15240)$ design or the unique $2$-$(176,160,19080)$ design.