  In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of
$Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here,
we determine for which of these groups the restriction of the metaplectic
representation gives rise to a reproducing formula. In all the positive cases
we characterize the admissible vectors with a generalized Calder\'on equation.
They include products of 1D-wavelets, directional wavelets, shearlets, and many
new examples.
