Cohomology of symplectic groups and Meyer's signature theorem

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 $H^2(\mathsf{S}\mathsf{p}(2g,\mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to $\mathbb{Z}$ \to \widetilde{$\mathsf{S}\mathsf{p}$(2g,$\mathbb{Z}$)} \to $\mathsf{S}\mathsf{p}$(2g,$\mathbb{Z}$) \to 1$thepullbackoftheuniversalcoverof$$\mathsf{S}\mathsf{p}$(2g,$\mathbb{R}$)$, Deligne proved that every finite index subgroup of $\widetilde{\mathsf {Sp}(2g, $\mathbb{Z}$)}$contains$2$\mathbb{Z}$$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{S}\mathsf{p}(2g,\mathbb{Z})$ 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 $\mathsf{S}\mathsf{p}(2g,\mathbb{Z})$ that contains this information. This quotient $\mathfrak{H}$ is a non-split extension of $\mathsf{S}\mathsf{p}(2g,2)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}\mathrm{/}2\to \tilde{\mathfrak{H}}\to \mathfrak{H}\to 1$, and $\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover $\mathsf{M}\mathsf{p}(2g,\mathbb{Z})=\widetilde{\mathsf{S}\mathsf{p}(2g,\mathbb{Z})}\mathrm{/}2\mathbb{Z}$. It is an extension of $\mathsf{S}\mathsf{p}(2g,2)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g\ge 4$, $\tilde{\mathfrak{H}}$ is the universal central extension of $\mathfrak{H}$. Putting all this together, we provide a recipe for computing the signature modulo $8$, and indicate some consequences.