The impact of the solubilizer of an element on the structure of a finite group

Let $G$ be a finite group, and let $x$ be an element of $G$. Denote by ${\text{Sol}}_G(x)$ the set of all $y\in G$ such that the group generated by $x$ and $y$ is soluble. We investigate the influence of ${\text{Sol}}_G(x)$ on the structure of $G$.