  We give an identification between the planar algebra of the
subgroup-subfactor $R \rtimes H \subset R \rtimes G$ and the $G$-invariant
planar subalgebra of the planar algebra of the bipartite graph $\star_n$, where
$n = [G : H]$. The crucial step in this identification is an exhibition of a
model for the basic construction tower, and thereafter of the standard
invariant, of $R \rtimes H \subset R \rtimes G$ in terms of operator matrices.
We also obtain an identification between the planar algebra of the fixed
algebra subfactor $R^G \subset R^H$ and the $G$-invariant planar subalgebra of
the planar algebra of the `flip' of $\star_n $.
