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".

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".

Added: Concerning your "hope" expressed in the comment above: If $A=k$ is a field (with trivial $G$-action) and $H$ is central, then the $E_2$-term becomes $$E_2^{\ast,\ast}=H^\ast_{G/H}(M;k)\otimes_k H^\ast(H;k).$$

By taking $G$ finite and $M$ a point it's obvious that one can't hope for more in general.