Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and $f'': G' \rightarrow G$(a Lie group morphism) with $f'(u'a')=f'(u')f''(a'), \forall u'\in P', a'\in G'.$(Cf. Kobayashi & Nomizu, volume 1, p53.) Let $\omega$ be a connection (1-form with values in the Lie algebra of $G$ that is $ad$-invariant) on the bundle $P \rightarrow X$.
I want to know: under what condition the connection $\omega$ on the bundle $P$ can be lifted to a connection $\omega'$ on $P'$, i.e. $f''_*(\omega')=\omega$? Moreover, if this happens, then under what condition $\omega'$ is uniquely determined by $\omega$?
In Kobayashi & Nomizu, volume 1, pp79~83, there are some theorems concerning mappings of connections, but there is no one concern my question.
It seems to be true when both the maps $(f',f'')$ are covering maps, since in the book "Spin Geometry" by Lawson and Michelsohn, page 108, before proposition 4.ll, the authors seem to assume this kind of conclusion. I prefer the "most general" conditions!