  A generalization of highly symmetric frames is presented by considering also
projective stabilizers of frame vectors. This allows construction of highly
symmetric line systems and study of highly symmetric frames in a more unified
manner. Construction of highly symmetric line systems involves computation of
twisted spherical functions associated with finite groups. Further
generalizations include definition of highly symmetric systems of subspaces. We
give several examples which illustrate our approach including 3 new kissing
configurations which improve lower bounds on the kissing number in $d=10,11,14$
to 510, 592 and 1932 respectively.
