The classification of non-solvable transitive BLT-sets of $\mathcal{Q}(4,q)$

For an odd prime power $q$, a BLT-set $\mathcal{B}$ is a set of $(q+1)$ points of $\mathcal{Q}(4,q)$ such that every point of $\mathcal{Q}(4,q)$ is collinear with at most two points of $\mathcal{B}$. We complete the classification of BLT-sets of $\mathcal{Q}(4,q)$ admitting a non-solvable transitive automorphism group. Nelson and Penttila completed the classification of BLT-sets $\mathcal{B}$ with a non-solvable transitive automorphism group acting irreducibly on the underlying $5$-dimensional vector space. In this paper, we prove that a BLT-set with a non-solvable transitive full automorphism group acting reducibly on the underlying $5$-dimensional vector space is either a classical BLT-set or a Kantor semifield BLT-set.