Base sizes for primitive groups with soluble stabilisers

Let $G$ be a finite primitive permutation group on set $\Omega$ with point stabiliser $H$. Recall that subset of is base for if its pointwise trivial. We define the size $G$, denoted $b(G,H)$, to minimal $G$. Determining fundamental problem in theory, long history stretching back 19th century. Here one our main motivations theorem Seress from 1996, which states $b(G,H) \leqslant 4$ soluble. In this paper we extend Seress' result by proving 5$ all groups soluble This bound best possible. also determine exact almost simple and study random bases setting. For example, prove probability $4$ elements form tends $1$ as $|G|$ infinity.