  We construct the vortex Floer homology group $VHF (M,\mu;H)$ for an
aspherical Hamiltonian $G$-manifold $(M, \omega)$ with moment map $\mu$ and a
class of $G$-invariant Hamiltonian loop $H_t$, following the proposal of [3].
This is a substitute for the ordinary Hamiltonian Floer homology of the
symplectic quotient of $M$. We achieve the transversality of the moduli space
by the classical perturbation argument instead of the virtual technique, so the
homology can be defined over ${\mathbb Z}$ or ${\mathbb Z}_2$.
