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As mentioned previously, we can define Hn(G,Z) = Hn(B(G),Z) where B(G) is any CW-space with fundamental group equal to G and for which all other homotopy groups are trivial. Given a short exact sequence of groups 1 → N → G → Q → 1 we set B(G,N) equal to the cofibre of the induced cofibration B(G) → B(Q), and we define

Hn(G,N,Z) = Hn+1(B(G,N),Z)

for all n>0. The homology exact sequence of the cofibration can then be written as

··· → H3(Q,Z) → H2(G,N,Z) → H2(G,Z) → H2(Q,Z) → H1(G,N,Z) → H1(G,Z) → H1(Q,Z) → 0 .

There is an isomorphism

H1(G,N,Z) = N/[N,G]

and textbooks often refer to the first five terms of the cofibration exact sequence, with third term replaced by N/[N,G], as the five-term Hochschild-Serre exact sequence (since these five terms can also be derived from the Hochschild-Serre spectral sequence for group extensions). Less well-known is that, in light of an isomorphism

H2(G,N,Z) = Ker( N ^ G → N ) ,

the first eight terms of the cofibration sequence are in fact a useful computational tool. The isomorphism for H2(G,N,Z) involves a nonabelian exterior product (a quotient of the nonabelian tensor product of the previous page) and was proved by a topological argument in [R. Brown & J.-L. Loday, "van Kampen theorems diagrams for diagrams of spaces", Topology 1987] and by an algebraic argument in [G. Ellis, "Nonabelian exterior products of groups and an exact sequence in the homology of groups", Glasgow Math. J. 29 (1987), 13-19].
For a finite group G we refer to the homology group H2(G,N,Z) as the relative Schur multiplier. When N=G this is the usual Schur multiplier H2(G,G,Z) = H2(G,Z). The following commands show that, for G the Sylow 2-subgroup of the Mathieu group M24 and N its commutator subgroup, the relative Schur multiplier is H2(G,N,Z) = (Z2)12 .

gap> G:=SylowSubgroup(MathieuGroup(24),2);; gap> N:=DerivedSubgroup(G);; gap> RelativeSchurMultiplier(G,N); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

For a finite group G the Universal Coefficient Theorem implies an isomorphism H2(G,Z) = H2(G,C×) where C× is the group of non-zero complex numbers. The group H2(G,C×) first appeared in work of Schur on complex projective representations G → PGL(C). He proved, for example, that every projective representation of G lifts to a linear representation G → GL(C) if and only if H2(G,C×) = 0.
The relative Schur multiplier has a similar interpretation (though it does not seem to be recorded anywhere in the literature). Let p:GL(C) → PGL(C) be the canonical projection and note that Ker(p) is central in GL(C). Suppose that N is normal in G and that we have a homomorphism f:G → PGL(C). Let us say that a homomorphism h:N → GL(C) is a relative lift of f if:

- ph( x )=f( x ) for all x in N,
- h( gxg-1 )= g' h( x ) g'-1 for all x in N and all g' in GL(C), g in G satisfying p( g' ) = f ( g ).

Every projective representation f:G → PGL(C) admits a relative lift h:N → GL(C) if and only if H2(G,N,Z) = 0. (At least, I think this is the correct statement!)

A second application of the (relative) Schur multiplier concerns groups G that are isomorphic to a quotient G = K/Z(K) of a group K by the centre of K. Such groups G are said to be capable. Beyl, Felgner and Schmid showed that, using the Schur multiplier, one can define a characteristic subgroup Z*(G) of the centre of G with the property that Z*(G)=0 if and only if G is capable. For details, see the paper [F.R. Beyl, U. Felgner and P. Schmid, "On groups occuring as central factor groups", J. Algebra 61 (1979), 161-177] . The group Z*(G) has recently become known as the epicentre of G.
More generally, given a normal subgroup N in G, a relative central extension of the pair (G,N) consists of a group homomorphism d:M → G and action (g,m) → gm of G on M satisfying:

1. d(gm) = gd( m )g-1 for g in G and m in M;
2. m m' m-1 = d(m) m for m and m' in M;
3. N = Image( d) ;
4. the action of G on M is such that G acts trivially on the kernel of d.

(Conditions 1 and 2 assert that d:M → G is a crossed module.) The pair (G,N) is said to be capable if it admits a relative central extension with the property that Ker( d ) consists precisely of those elements in M on which G acts trivially. Using the relative Schur multiplier one can define a subgroup Z*(G,N) of the centre of N with the property that Z*(G,N) = 0 if and only if the pair (G,N) is capable. When N=G the group Z*(G,G) coincides with the epicenter Z*(G) of Beyl, Felgner and Schmid.
The following commands show that, for G the sylow 2-subgroup of the Mathieu group M24 and N equal to the centre of G, the pair (G,N) is capable. They also show that the group G itself is not capable.

gap> G:=SylowSubgroup(MathieuGroup(24),2);; gap> N:=Centre(G);; gap> Order(Epicentre(G,N)); 1 gap> Order(Epicentre(G)); 2

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