The main result of this paper is that any $3$-dimensional manifold with a finite group action is equivariantly, invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology $3$-sphere also acts on a hyperbolic homology $3$-sphere. The theorem has other applications, including establishing the existence of an infinite number of hyperbolic homology spheres with a free $Z_p$ action that does not extend to any contractible manifold. A non-equivariant version yields an infinite number of hyperbolic integer homology spheres that bound integer homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on $3$-manifolds is antisymmetric.