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We prove that, in the case $|P|\\geq 8$, $TT(G)\\cong X(G)$ the group of one-dimensional $kG$-modules, except possibly when $G/O_{2'}(G)\\cong \\mathfrak{A}_6$, the alternating group of degree $6$; in which case $G$ may have $9$-dimensional simple torsion endo-trivial modules. 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