There are plenty of such manifolds, but as Danny indicates in his answer, there is not a known classification.
Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skeleton $C^{(1)}$ of $C$ is a wedge of $k$ circles. Then take a 4-dimensional handlebody $H$ with $k$ 1-handles, with a spine of $C^{(1)}$. There are $k$ disks attached to the 1-skeleton in $C$. Attach $2$-handles to $H$ in such a way that the core of the attaching map is homotopic to the attaching map in the 2-skeleton $C^{(1)}$ to get a manifold $W$ with [handle structure][1] so that $C$ is a deformation retract of $W$, and hence $\pi_1(W)\cong G$. By the [Poincaré-Lefschetz theorem][2], $\partial W$ is a homology 3-sphere. But in general we may get many different boundaries depending on the choice of isotopy class and framing of the boundary of the cores of the 2-handles.
To get such groups $G$, one can choose a [small-cancellation][3]
$C'(\frac16)$ balanced presentation with $H_1(G)=0$. Then a presentation complex $C$ will be aspherical and acyclic. ***Added:*** See an explicit example due to Rylee Lyman in the comments. A simpler presentation of the Higman group is given (which perfect and has aspherical presentation complex).
The difficulty here is that one has no idea what 3-manifold the boundary of such a manifold will be. Moreover, it's not clear what the homeomorphism classification of such manifolds is, even if they have the same aspherical 2-skeleton spine and boundary.
Presumably there are also examples which do not have a 2-dimensional spine. The only obvious restriction I see is that the fundamental group must have cohomological dimension three.
[1]: https://en.wikipedia.org/wiki/Handle_decomposition
[2]: https://en.wikipedia.org/wiki/Lefschetz_duality
[3]: https://en.wikipedia.org/wiki/Small_cancellation_theory#Asphericity
[4]: https://people.math.osu.edu/davis.12/pdgroup.pdf