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.