A Theorem of Siemons and Wagner

In 1988 Siemons and Wagner describe a relationship between the lengths of $G$-orbits on subsets of a $G$-set $\mathrm{\Omega }$. They highlighted the situation where $\mathrm{\Delta }\subset \mathrm{\Omega }$ with $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }=k,$ and $\mathrm{\mid }{\mathrm{\Delta }}^G\mathrm{\mid }>\mathrm{\mid }{\mathrm{\Sigma }}^G\mathrm{\mid }$ for all $(k+1)$-subsets, $\mathrm{\Sigma }$ of $\mathrm{\Omega }$ where $\mathrm{\Delta }\subset \mathrm{\Sigma }$. They went on to classify all primitive groups with this property for $k=2$. Here we address some questions about primitive permutation groups satisfying this property when $k=3$ and list all $3$-homogeneous groups satisfying this condition.