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We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$is either a Lubin-Tate spectrum $E_n$ or the related 2-periodic Morava K-theory$K_n$, and $G$ is a finite group. When $R$ is an Eilenberg-Mac Lane spectrum,in some good cases such an extension is a $G$-Galois extension in the sense ofJohn Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ theseextensions are always faithful in the $K_n$ local category. However, for acyclic $p$-group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \toF({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it ramifies. As aconsequence, it follows that the $E_n$-theory Eilenberg-Moore spectral sequencefor $G$ and $BG$ does not always converge to its expected target.

Some properties of Lubin-Tate cohomology for classifying spaces of finite groups