Groups Obtained from $2-(n,4,3)$ Supersimple Designs

We contribute towards the classification programme for Conway groupoids associated to a $2-(n,4,\lambda)$ design. Our main results improve the known bounds for a hole stabilizer to be primitive, or to contain the alternating group, ${\rm Alt}(n-1)$. We exploit these improved bounds to give a partial classification for Conway groupoids when $\lambda=3$.