We say that a $G$-module $I$ is induced if $$I\cong L\otimes\mathbb{Z}G$$ where $L$ is an abelian group and the action on $L\otimes\mathbb{Z}G$ is given by the action of $G$ only on the second component, so that $$g(l\otimes h)=l\otimes gh$$ Here comes my question: is it true that if $$H^k(G,\mathbb{Z}G)=0$$ then $$H^k(G,I)=0$$ for any induced $G$-module $I$?
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