Ballester-Bolinches and Guo showed that a finite group 𝐺 is 2-nilpotent if 𝐺 satisfies: (1) a Sylow 2-subgroup 𝑃 of 𝐺 is quaternion-free and (2) $\Omega_1(P\cap G')\leq Z(P)$ and $N_G(P)$ is 2-nilpotent. In this paper, it is obtained that 𝐺 is a non-2-nilpotent group of order $16q$ for an odd prime 𝑞 satisfying (1) a Sylow 2-subgroup 𝑃 of 𝐺 is not quaternion-free and (2) $\Omega_1(P\cap G')\leq Z(P)$ and $N_G(P)$ is 2-nilpotent if and only if $q=3$ and $G\cong GL_2(3)$.
