  Given a complex vector space $V$ of finite dimension, its Grassmannian
variety parametrizes all subspaces of $V$ of a given dimension. Similarly, if a
finite group $G$ acts on $V$, its invariant Grassmannian parametrizes all the
$G$-invariant subspaces of $V$ of a given dimension. Based on this fact, we
develop an algorithm for computing $G$-invariant projective varieties arising
as an intersection of hypersurfaces of the same degree.
  We apply the algorithm to find a projective model of a polarized K3 surface
with a faithful action of $T_{192}\rtimes \mu_2$ and some further symmetric K3
surfaces with a degree 8 polarization.
