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To elaborate on Eric's answer, I believe that H1-n(G, A) is πn of the homotopy fixed point space K(A, 1)hG. That exact sequence which ends at H1--which is only a set, while H0 is a group--is actually the long exact sequence of the fibration K(A, 1)hG -> K(B, 1)hG -> K(C, 1)hG in disguise.

That reindexing 1-n is related to the 1 in K(A, 1); if A is abelian, then we can replace all the occurrences of 1 by r for any r ≥ 0, giving arbitrarily long segments of the long exact sequence. (Or you can use the language of spectra.spectra: H-n(G, A) = πn((HA)hG). (HA)hG has nonzero homotopy groups only in non*positive* dimension.)

K(A, 1)hG is a groupoid (it only has homotopy in dimensions 0 and 1) and I expect it should be the groupoid of some kind of extensions of G by A (where morphisms are isomorphisms of extensions which are the identity on G and A). H1(G, A) would then classify those extensions, and H0(G, A) = AG would be the automorphism group of the basepoint, which I guess is the semidirect product (this is easy to check).
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To elaborate on Eric's answer, I believe that H1-n(G, A) is πn of the homotopy fixed point space K(A, 1)hG. That exact sequence which ends at H1--which is only a set, while H0 is a group--is actually the long exact sequence of the fibration K(A, 1)hG -> K(B, 1)hG -> K(C, 1)hG in disguise.

That reindexing 1-n is related to the 1 in K(A, 1); if A is abelian, then we can replace all the occurrences of 1 by r for any r ≥ 0, giving arbitrarily long segments of the long exact sequence. (Or you can use the language of spectra.)

K(A, 1)hG is a groupoid (it only has homotopy in dimensions 0 and 1) and I expect it should be the groupoid of some kind of extensions of G by A (where morphisms are isomorphisms of extensions which are the identity on G and A). H1(G, A) would then classify those extensions, and H0(G, A) = AG would be the automorphism group of the basepoint, which I guess is the semidirect product (this is easy to check).
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To elaborate on Eric's answer, I believe that H1-n(G, A) is πn of the homotopy fixed point space K(A, 1)hG. That exact sequence which ends at H1--which is only a set, while H0 is a group--is actually the long exact sequence of the fibration K(A, 1)hG -> K(B, 1)hG -> K(C, 1)hG in disguise.

That reindexing 1-n is related to the 1 in K(A, 1); if A is abelian, then we can replace all the occurrences of 1 by r for any r ≥ 0, giving arbitrarily long segments of the long exact sequence. (Or you can use the language of spectra.)