  We deal with the Weak Lefschetz property (WLP) for Artinian standard graded
Gorenstein algebras of codimension $3.$ We prove that many Gorenstein sequences
force the WLP for such algebras. Moreover for every Gorenstein sequence $H$ of
codimension 3 we found several Gorenstein Betti sequences compatible with $H$
which again force the WLP. Finally we show that for every Gorenstein Betti
sequence the general Artinian standard graded Gorenstein algebra with such
Betti sequence has the WLP.
