Geodesic
On finite groups with $Q$-central elements of prime order
Trudy Instituta matematiki, Tome 16 (2008) no. 1, pp. 97-99.

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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 $G$. 
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O. L. Shemetkova. On finite groups with $Q$-central elements of prime order. Trudy Instituta matematiki, Tome 16 (2008) no. 1, pp. 97-99. http://geodesic.mathdoc.fr/item/TIMB_2008_16_1_a16/

[1] Shemetkov L.A., “A new concept of a generalized central element.”, Mezhdunar. algebraich. konf. “Klassy grupp i algebr”, posvyaschennaya 100-letiyu so dnya rozhdeniya S.A. Chunikhina, Tez. dokl., Gomelskii gosuniverstet im. F. Skoriny, Gomel, 2005, 22–23

[2] Ito N., “Note on $(LM)$-groups of finite orders”, Kodai Math. Semin. Rep., 1–2 (1951), 1–6 | DOI | MR | Zbl