Simulations of an Ising q-state Pods model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for q=2, there are two critical Ising lines meeting at T=0 at the four-state Potts critical point; (ii) for 24, the Ising:critical line intersects a Line of first-order transitions at a critical end point.