  We consider a class of minimal abelian extensions of the Standard Model with
an extra neutral gauge boson $Z'$ at the TeV scale. In these scenarios an
extended scalar sector and heavy right-handed neutrinos are naturally
envisaged. We present some of their striking signatures at the Large Hadron
Collider, the most interesting arising from a $Z'$ decaying to heavy neutrino
pairs as well as a heavy scalar decaying to two Standard Model Higgses. Using
renormalisation group methods, we characterise the high energy behaviours of
these extensions and exploit the constraints imposed by the embedding into a
wider GUT scenario.
