  We study lifting problems for operator semigroups in the Calkin algebra
$\mathscr{Q}(\mathcal{H})$, our approach being mainly based on the
Brown--Douglas--Fillmore theory. With any normal $C_0$-semigroup $(q(t))_{t\geq
0}$ in $\mathscr{Q}(\mathcal{H})$ we associate an extension
$\Gamma\in\mathrm{Ext}(\Delta)$, where $\Delta$ is the inverse limit of certain
compact metric spaces defined purely in terms of the spectrum $\sigma(A)$ of
the generator of $(q(t))_{t\geq 0}$. By using Milnor's exact sequence, we show
that if each $q(t)$ has a normal lift, then the question whether $\Gamma$ is
trivial reduces to the question whether the corresponding first derived functor
vanishes. With the aid of the CRISP property and Kasparov's Technical Theorem,
we provide geometric conditions on $\sigma(A)$ which guarantee splitting of
$\Gamma$. If $\Delta$ is a perfect compact metric space, we obtain in this way
a $C_0$-semigroup $(Q(t))_{t\geq 0}$ which lifts $(q(t))_{t\geq 0}$ on dyadic
rationals.
