Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denote the conjugacy class of $x$ in $G$. Is it true that $|C_G(x):Z(G)|=2$?

p.s.
I know that if $|G|\leq 2^8$ then the equality holds.