A complete classification of the $(15_4 20_3)$-configurations with at least three $K_5$-graphs

The class of $\left(\binom{n+1}{2}_{n-1} \binom{n+1}{3}_3\right)$-configurations which contain at least $n-2$ $K_n$-graphs coincides with the class of so called systems of triangle perspectives i.e. of configurations which contain a bundle of $n-2$ Pasch configurations with a common line. For $n=5$ the class consists of all binomial partial Steiner triple systems on $15$ points, that contain at least three $K_5$-graphs. In this case a complete classification of respective configurations is given and their automorphisms are determined.