  Let $G$ be a unimodular type I second countable locally compact group and
$\hat G$ its unitary dual. We introduce and study a global pseudo-differential
calculus for operator-valued symbols defined on $G\times\hat G$, and its
relations to suitably defined Wigner transforms and Weyl systems. We also
unveil its connections with crossed products $C^*$-algebras associated to
certain $C^*$-dynamical systems, and apply it to the spectral analysis of
covariant families of operators. Applications are given to nilpotent Lie
groups, in which case we relate quantizations with operator-valued and
scalar-valued symbols.
