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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