Given a pair of finite groups H ⊂ G, and an outer action α of G on the hyperfinite II1- factor R, we give an identification between the planar algebra of the subgroup-subfactor R ⋊α/H H ⊂ R ⋊α G and the G-invariant planar subalgebra of the planar algebra of the bipartite graph ⋆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 ⋊α/H H ⊂ Rα ⋊ G in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor RG ⊂ RH and the G-invariant planar subalgebra of the planar algebra of the ‘flip’ of ⋆n.