  In this paper, we study the properties of the scaled packing topological
pressures for topological dynamical system $(X,G)$, where $G$ is a countable
discrete infinite amenable group. We show that the scaled packing topological
pressures can be determined by the scaled Bowen topological pressures. We
obtain Billingsley's Theorem for the scaled packing pressures with a
$G$-action. Then we get a variational principle between the scaled packing
pressures and the scaled measure-theoretic upper local pressures. Finally, we
give some restrictions on the scaled sequence $\mathbf{b}$, then in the case of
the set $X_{\mu}$ of generic points, we prove that
  $$P^{P}(X_{\mu},\left\{F_{n}\right\},f,\mathbf{b})=h_{\mu}(X)+\int_{X} f
\mathrm{d}\mu,$$
  if $\left\{F_{n}\right\}$ is tempered and $\mu$ is a $G$-invariant ergodic
Borel probability measure.
