This is a very general question-- I'm really after any references to anything similar...

Let $A$ be a $C^*$-algebra equipped with a continuous one-parameter group of automorphisms $(\alpha_t)_{t\in\mathbb R}$ and let $\pi:A\rightarrow B(H)$ be a (non-degenerate) *-representation. I'm after conditions which give a $\sigma$-weakly continuous one-parameter group of automorphisms of $B(H)$, say $(\beta_t)$, such that $\pi\circ\alpha_t = \beta_t\circ\pi$.

As $\beta_t$ will be implemented as $\beta_t(x) = U_t x U_t^*$ where $(U_t)$ is a continuous group of unitaries (that is, a continuous unitary representation of $\mathbb R$). So I'm exactly asking:

When does $\pi$ arise as part of a

covariantrepresentation of $(A,\alpha)$?

As covariant representations biject with representations of the crossed product $A \rtimes_\alpha \mathbb R$, it seems that *in principle* knowing $A \rtimes_\alpha \mathbb R$ should be enough. But this seems terribly abstract!