I think you're looking for the Hochschild-Serre spectral sequence. It's slightly more complicated than a long exact sequence, but you can extract the very concrete "inflation-restriction sequence" out of it.

Your general question is a little too general to get a good answer, though there are some good other questions on this site that you will probably find very helpful, e.g.,

Intuition for Group Cohomology

Essential theorems in group (co)homology

If you really only want $n=0$ and $n=1$, these are very concrete and addressed in any of the standard references for group cohomology. $H^0$ is the fixed-point functor, and $H^1$ is the group of "crossed homomorphisms" (which reduce to regular homomorphisms when the action is trivial).

I think you're looking for the Hochschild-Serre spectral sequence. It's slightly more complicated than a long exact sequence, but you can extract the very concrete "inflation-restriction sequence" out of it.

Your general question is a little too general to get a good answer, though there are some good other questions on this site that you will probably find very helpful, e.g.,

Intuition for Group Cohomology

Essential theorems in group (co)homology

If you really only want $n=0$ and $n=1$, these are very concrete and addressed in any of the standard references for group cohomology. $H^0$ is the fixed-point functor, and $H^1$ is the group of "crossed homomorphisms" (which reduce to regular homomorphisms when the action is trivial).

I think you're looking for the Hochschild-Serre spectral sequence. It's slightly more complicated than a long exact sequence, but you can extract the very concrete "inflation-restriction sequence" out of it.

Your general question is a little too general to get a good answer, though there are some good other questions on this site that you will probably find very helpful, e.g.,

Intuition for Group Cohomology

Essential theorems in group (co)homology

I think you're looking for the Hochschild-Serre spectral sequence. It's slightly more complicated than a long exact sequence, but you can extract the very concrete "inflation-restriction sequence" out of it.

Your general question is a little too general to get a good answer, though there are some good other questions on this site that you will probably find very helpful, e.g.,

Intuition for Group Cohomology

Essential theorems in group (co)homology

If you really only want $n=0$ and $n=1$, these are very concrete and addressed in any of the standard references for group cohomology. $H^0$ is the fixed-point functor, and $H^1$ is the group of "crossed homomorphisms" (which reduce to regular homomorphisms when the action is trivial).