Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of 4, and can be computed using an element of
\[ H^2(\mathrm{Sp}(2g,\mathbb{Z});\mathbb{Z}) .\]
If we denote by
\[ 1 \to \mathbb{Z} \to \widetilde{\mathrm{Sp}(2g,\mathbb{Z})} \to \mathrm{Sp}(2g,\mathbb{Z}) \to 1 \]
the pullback of the universal cover of \( \mathrm{Sp}(2g,\mathbb{Z}) \), then by a theorem of Deligne, every finite index subgroup of \( \widetilde{\mathrm{Sp}(2g,\mathbb{Z})} \) contains \( 2\mathbb{Z} \). As a consequence, a class in the second cohomology of any finite quotient of \( \mathrm{Sp}(2g,\mathbb{Z}) \) can at most enable us to compute the signature of a surface bundle modulo 8. We show that this is in fact possible and investigate the smallest quotient of \( \mathrm{Sp}(2g,\mathbb{Z}) \) that contains this information. This quotient \( \mathfrak{h} \) is a nonsplit extension of \( \mathrm{Sp}(2g,2) \) by an elementary abelian group of order \( 2^{2g+1} \). There is a central extension
\[ 1 \to \mathbb{Z}/2 \to \tilde{\mathfrak{h}} \to \mathfrak{h} \to 1 ,\]
and \( \tilde{\mathfrak{h}} \) appears as a quotient of the metaplectic double cover
\[ \mathrm{Mp}(2g,\mathbb{Z}) = \widetilde{\mathrm{Sp}(2g,\mathbb{Z})}/2\mathbb{Z} .\]
It is an extension of \( \mathrm{Sp}(2g,2) \) by an almost extraspecial group of order \( 2^{2g+2} \), and has a faithful irreducible complex representation of dimension \( 2^g \) Provided \( g\geq 4 \), the extension \( \tilde{\mathfrak{h}} \) is the universal central extension of \( \mathfrak{h} \). Putting all this together, in Section 4 we provide a recipe for computing the signature modulo 8, and indicate some consequences.
