  In this paper we consider the wave equations for hypoelliptic homogeneous
left-invariant operators on graded Lie groups with time-dependent H\"older
propagation speeds. The examples are the time-dependent wave equation for the
sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or
$p$-evolution equations for higher order operators, already in all these cases
our results being new. We establish sharp well-posedness results in the spirit
of the classical result by Colombini, de Giorgi and Spagnolo. In particular, we
describe an interesting loss of regularity phenomenon depending on the step of
the group and on the order of the considered operator.
