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Let $G$ be a finite group with a Sylow 2-subgroup $P$ which is either quaternion or semi-dihedral. Let $k$ be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial $kG$-modules, whose restrictions to $P$ are isomorphic to the direct sum of the known exotic endotrivial $kP$-modules and some projective modules. This provides a description of the group $T\left(G\right)$ of endotrivial $kG$-modules.