Complex sentence: So any automorphism of the Heisenberg group, fixing the center, acts on this representation "H"--to be precise, the action is only well-defined up to multiplication by a non-zero constant.
Split into simpler sentences: So any automorphism of the Heisenberg group acts on this representation "H". However if we look more closely we notice that the action of an automorphism of the Heisenberg group on "H" is not uniquely defined, but only defined up to multiplication by a non-zero constant.