If $G$ is compact and $H$ is a closed normal subgroup of $G$ that acts trivially on $M$, then there is a spectral sequence $$E_2^{i,j}=H_{G/H}^i(M;H^j(H;A)) \Rightarrow H^{i+j}_G(M;A)$$ where $A$ is a $G$-module.
If $H$ is central in $G$ and $A$ is a trivial $G$-module then all coefficients are trivial. Otherwise they are understood to be local coefficients.
This is a result of Duflot and can be found in section 3 of her celebrated paper "Depth and equivariant cohomology".