Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup 
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*Is there an analogue of the Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup?

*What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$?

*What is the best technique to get the spectral sequence? For me the Grothendieck spectral sequence us much better than the spectral sequence of a filtered complex.

There is a parallel question which is likely easier.

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*Is there an analogue of the Hochschild–Serre spectral sequence for a Lie subalgebra which is not an ideal?

2 and 3 remain the same.
I already asked a version of this question on MathSE but got no responses.
 A: Sorry for reviving an old question, but it seems that the Kropholler spectral sequence exactly answers the first 3 questions:
Kropholler, P.H., A generalization of the Lyndon-Hochschild-Serre spectral sequence with applications to group cohomology and decompositions of groups., J. Group Theory 9, No. 1, 1-25 (2006). ZBL1115.20042.
A: I don't think so. The LHS spectral sequence can be thought of as the Serre spectral sequence associated to the fiber sequence
$$BN \to BG \to B(G/N)$$
where $G$ is a group and $N$ is a normal subgroup of it. If $N$ is not required to be normal then the third term in this fiber sequence no longer exists, so it's unclear to me in what sense we can have a reasonable analogue of the LHS spectral sequence here. 
A: For Lie algebras $\mathfrak h \subseteq \mathfrak g$ and a $\mathfrak g$-module $M$ there is a spectral sequence arising from the following filtration $F_i C^q(\mathfrak g, M)$ on the Chevalley-Eilenberg complex $C^q(\mathfrak g,M)$. The filtration is defined by
$$ F_iC^q(\mathfrak g, M) = \{ \gamma \mid i_{h_1}i_{h_2}\cdots i_{h_{q+1-i}}\gamma = 0 \text{ for all }h_i \in \mathfrak h\},$$
where $i_{h}$ means contraction by $h$ (putting $h$ into one argument).
The $E_1$-page may be identified as
$$ E_1^{j,i} = H^i(\mathfrak h, Hom(\wedge^j \mathfrak g/\mathfrak h, M)).$$
When $\mathfrak h$ is an ideal, all of this coincides with the Grothendieck spectral sequence for the composition of $(-)^{\mathfrak h}$ and $(-)^{\mathfrak g/\mathfrak h}$.

Reference: Hochschild, G., & Serre, J. P. (1953). Cohomology of Lie algebras. Annals of Mathematics, 591-603.
