  For a locally compact group $G$ and a strongly self-absorbing $G$-algebra
$(\mathcal{D},\delta)$, we obtain a new characterization of absorption of a
strongly self-absorbing action using almost equivariant completely positive
maps into the underlying algebra. The main technical tool to obtain this
characterization is the existence of almost equivariant lifts for equivariant
completely positive maps, proved in recent work of the authors.
  This characterization is then used to show that an equivariant
$C_0(X)$-algebra with $\mathrm{dim}_{\mathrm{cov}}(X)<\infty$ is
$(\mathcal{D},\delta)$-stable if and only if all of its fibers are, extending a
result of Hirshberg, R{\o}rdam and Winter to the equivariant setting. The
condition on the dimension of $X$ is known to be necessary, and we show that it
can be removed if, for example, the bundle is locally trivial.
