I am interested in knowing if there is a closed, (smooth) aspherical manifold $M$ (hyperbolic would be best) with superperfect fundamental group (that is to say, with $H_1(\pi_1(M);\mathbb{Z}) = H_2(\pi_1(M);\mathbb{Z}) = 0$; note $H_1(\pi_1(M);\mathbb{Z}) = 0$ is equivalent to perfect) and non-trivial center ($\mathbb{Z}_2$ would be best, but any f.g. Abelian group will do).

Also, assuming there is a manifold that fits the criteria, I would likely need a handlebody decomposition for the manifold, assuming the "standard handlebody procedure" for producing a closed, smooth manifold from a prescribed finite presentation of a/the fundamental group does not yield the smooth manifold in question (e.g., the "standard manifold" is not aspherical).

I found many hyperbolic 3-manifolds with superperfect fundamental group using SnapPy, but SnapPy evidently doesn't have a center "method" for the fundamental group method/class attached to 3-manifolds. Sage/GAP/MAGMA also appear not to be able to compute the center for an infinite finitely-presented fundamental group.

Thanks much in advance. I realize this is kind-of "shooting for the moon/stars".