  The classification problem for K3-surfaces equipped with finite groups $H$ of
symplectic symmetry centralized by an antisymplectic involution is considered.
An approach via equivariant Mori-reduction is employed. This method, which has
proved to be successful even for rather small groups, is exemplified here by
giving a complete classification in the case $H = C_3 \ltimes C_7$. The
consideration of this particular group is related to the study of K3-surfaces
with maximal finite groups of symplectic automorphisms. Applications to the
case $L_2(7)$ are given.
