  We first study the existence, uniqueness and well-posedness of a general
class of integro-differential diffusion equation on $L^p(\mathbb{G})$
$(1<p<+\infty$, $\mathbb{G}$ is a graded Lie group). We show the explicit
solution of the considered equation by using the Fourier analysis of the group.
The equation involves a nonlocal in time operator (with a general kernel) and a
positive Rockland operator acting on $\mathbb{G}.$ Also, we provide
$L^p(\mathbb{G})-L^q(\mathbb{G})$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm
estimates and time decay rate for the solutions. In fact, by using some
contemporary results, one can translate the latter regularity problem to the
study of boundedness of its propagator which strongly depends on the traces of
the spectral projections of the Rockland operator. Moreover, in many cases, we
can obtain time asymptotic decay for the solutions which depends intrinsically
on the considered kernel. As a complement, we give some norm estimates for the
solutions in terms of a homogeneous Sobolev space in $L^{2}(\mathbb{G})$ that
involves the Rockland operator. We also give a counterpart of our results in
the setting of compact Lie groups. Illustrative examples are also given.
