Packing of regular tetrahedral quartets of circles on a sphere

How must 4N non-overlapping equal circles forming N quartets be packed on a sphere so that the angular diameter of the circles will be as large as possible under the constraint that, within each quartet, the circle centres lie at the vertices of a regular tetrahedron? Computer-generated solutions to this optimization problem are presented for N = 1-8 quartets. A check of Danzerian rigidity of the computed packings is made in each case. The configurations so obtained are characterized as compounds of regular tetrahedra, some well known and some newly defined.