  Given any topological group $G$, the topological classification of principal
$G$-bundles over a finite CW-complex $X$ is long-known to be given by the set
of free homotopy classes of maps from $X$ to the corresponding classifying
space $BG$. This classical result has been long-used to provide such
classification in terms of explicit characteristic classes. However, even when
$X$ has dimension $2$, it seems there is a case in which such explicit
classification has not been explicitly considered. This is the case where $G$
is a Lie group, whose group of components acts non-trivially on its fundamental
group $\pi_1G$. In this note we deal with this case by obtaining the
classification, in terms of characteristic classes, of principal $G$-bundles
over a finite CW-complex of dimension $2$, with $G$ is a Lie group such that
$\pi_0G$ is abelian.
