  We generalize the BF theory action to the case of a general Lie crossed
module $(H \to G)$, where $G$ and $H$ are non-abelian Lie groups. Our
construction requires the existence of $G$-invariant non-degenerate bilinear
forms on the Lie algebras of $G$ and $H$ and we show that there are many
examples of such Lie crossed modules by using the construction of crossed
modules provided by short chain complexes of vector spaces. We also generalize
this construction to an arbitrary chain complex of vector spaces, of finite
type. We construct two gauge-invariant actions for 2-flat and fake-flat
2-connections with auxiliary fields. The first action is of the same type as
the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class
of Lie crossed modules, where $H$ is abelian. The second action is an extended
BFCG action which contains an additional auxiliary field. However, these two
actions are related by a field redefinition. We also construct a
three-parameter deformation of the extended BFCG action, which we believe to be
relevant for the construction of non-trivial invariants of knotted surfaces
embedded in the four-sphere.
