I privately asked Symonds about the matter and he seems to agree with me that the corollary in this form is wrong. My counterexample should be valid.

The point is that this corollary 4.5.2 is used in corollary 5.2.5 of the cited paper to prove that the right action on the dualizing module of a Poincarè group is given by the determinant of the adjoint action on L(G), the logarithm algebra associated to G. In the proof we actually show that that for a precise module $M$ we have $H^n(G;M)\otimes \mathbb{Q}_p \neq 0$ and we use the implication $1)\Rightarrow 2)$ to conclude.

Since we show the rationalization is not trivial, modifying $1)$ to ask $H^n(G;M)$ to be torsion-free or $\mathbb{Z}_p$ does not break the proof of corollary 5.2.5. I am not aware of any instance in the literature where this corollary 4.5.2 is used in this wrong form, if you know such a case maybe reporting it here could be useful to the discussion.

I privately asked Symonds about the matter and he seems to agree with me that the corollary in this form is wrong. My counterexample should be valid.

The point is that this corollary 4.5.2 is used in corollary 5.2.5 of the cited paper to prove that the right action on the dualizing module of a Poincarè group is given by the determinant of the adjoint action on $L(G)$, the logarithm algebra associated to $G$. In the proof we actually show that that for a precise module $M$ we have $H^n(G;M)\otimes \mathbb{Q}_p \neq 0$ and we use the implication $1)\Rightarrow 2)$ to conclude.

Since we show the rationalization is not trivial, modifying $1)$ to ask $H^n(G;M)$ to be torsion-free or $\mathbb{Z}_p$ does not break the proof of corollary 5.2.5. I am not aware of any instance in the literature where this corollary 4.5.2 is used in this wrong form, if you know such a case maybe reporting it here could be useful to the discussion.