  We study the phase-space concentration of the so-called generalized
metaplectic operators whose main examples are Schr\"odinger equations with
bounded perturbations.
  To reach this goal, we perform a so-called $\mathcal{A}$-Wigner analysis of
the previous equations, as started in Part I, cf. [14]. Namely, the classical
Wigner distribution is extended by considering a class of time-frequency
representations constructed as images of metaplectic operators acting on
symplectic matrices $\mathcal{A}\in Sp(2d,\mathbb{R})$. Sub-classes of these
representations, related to covariant symplectic matrices, reveal to be
particularly suited for the time-frequency study of the Schr\"odinger
evolution. This testifies the effectiveness of this approach for such
equations, highlighted by the development of a related wave front set.
  We first study the properties of $\mathcal{A}$-Wigner representations and
related pseudodifferential operators needed for our goal. This approach paves
the way to new quantization procedures.
  As a byproduct, we introduce new quasi-algebras of generalized metaplectic
operators containing Schr\"odinger equations with more general potentials,
extending the results contained in the previous works [8,9].
