Update: I managed to partially answer this question after posing it. For coefficient fields relatively prime to $n$!, there is an isomorphism $ H_G^*(M) \cong H^*(BLO(n))$ where $BLO(n)$ is the classifying space of the continuous loop group $LO(n)$, while in characteristic 2 there is an isomorphism $ H_G^*(M) \cong H^*(BLU(n))$.
Details are found in Corollary 6.4 of https://arxiv.org/abs/1312.7450.
