  We describe a flexible technique that constructs tight fusion frames with
prescribed transitive symmetry. Applying this technique with representations of
the symmetric and alternating groups, we obtain several new infinite families
of equi-isoclinic tight fusion frames, each with the remarkable property that
its automorphism group is either $S_n$ or $A_n$. These ensembles are optimal
packings for Grassmannian space equipped with spectral distance, and as such,
they find applications in block compressed sensing.
