  Inspired by an analogous result of Arnautov about isomorphisms, we prove that
all continuous surjective homomorphisms of topological groups f:G-->H can be
obtained as restrictions of open continuous surjective homomorphisms f':G'-->H,
where G is a topological subgroup of G'. In case the topologies on G and H are
Hausdorff and H is complete, we characterize continuous surjective
homomorphisms f:G-->H when G has to be a dense normal subgroup of G'. We
consider the general case when G is requested to be a normal subgroup of G',
that is when f is semitopological - Arnautov gave a characterization of
semitopological isomorphisms internal to the groups G and H. In the case of
homomorphisms we define new properties and consider particular cases in order
to give similar internal conditions which are sufficient or necessary for f to
be semitopological. Finally we establish a lot of stability properties of the
class of all semitopological homomorphisms.
