Central extensions of the symmetric core of a group (Q1324951)

If \(G\) is a group and \(\sigma\) is an involutory automorphism of \(G\), the symmetric core of \((G,\sigma)\) is \(S(G) := \{a \sigma(a^{-1}) \mid a\in G\}\). Let \((H,Z)\) be a central extension of \(G\) with exact sequence \(1 \to Z \to H @>\phi>> G \to 1\), where \(Z \subseteq Z(H)\) and let \(\tau\) be an involutory automorphism of \(H\) acting trivially on \(Z\) and such that \(\phi\tau = \sigma\phi\). In this case \((H,Z,\tau)\) is called a central extension (of the null type) of \((G,\sigma)\); since the symmetric core \(S(H)\) of \((H,\tau)\) maps homomorphically onto \(S(G)\), \(S(H)\) is called a central extension of \(S(G)\). The author constructs a ``generic cover'' \(S(U)\) of \(S(G)\), in the sense that every cover of \(S(G)\) is a homomorphic image of \(S(U)\) and proves that \(S(U)\) is finite and uniquely determined up to isomorphism if \(G\) is finite. If \(G\) is finite, it is shown that the index of \(S(G)\) in \(S(U)\) divides the order of the restricted Schur multiplier \(H^ 2(G,C;\sigma)\), where \(C\) is the multiplicative group of nonzero complex numbers.