On finite groups with $Q$-central elements of prime order

Following L. A. Shemetkov, an element $x$ of a non-nilpotent finite group $X$ is called a $Q$-central element if there exists a central chief factor $H/L$ of $X$ such that $x\in H$ and $x\notin L$. An element $x$ is called a $Q_8$-element in a group if there exists a section $A/B$ such that $A/B$ contains $xB$ and is isomorphic to the quaternion group $Q_8$ of order $8$, and $o(x)$ coincides with the order of $xB$ in $A/B$. Let $G$ be a finite group such that every its element of prime order is $Q$-central. Then the following conditions hold: 1) a Sylow 2-subgroup $G_2$ of $G$ is normal and $G/G_2$ is nilpotent; 2) there is a $Q_8$-element in $G_2$ which is not $Q$-central in $G$.