The Hénon family of planar maps is considered driven by the Arnold family of circle maps. This leads to a five-parameter family of skew product systems on the solid torus. In this paper the dynamics of this skew product family and its perturbations are studied. It is shown that, in certain parameter domains, Hénon-like strange attractors occur. The existence of quasi-periodic Hénon-like attractors is partially discussed, and further supported by numerical evidence.