On Solublizer and Normalizer Conjecture

Let $G$ be a finite group and $x$ be an element of $G$. Let $Sol_G(x)$ denote the set of all $y in G$ such that $cyc{x,y}$ is soluble. We find the structure $G$ when $Sol_G(x)$ is meta-cyclic maximal $2$-subgroup of $G$. Also, we provide an equivalent condition for the normalizer conjecture, that is $|N_G(cyc{x})|mid, |Sol_G(x)|$.