We study a projective unitary representation of the productG=G˜×G, whereGis a locally compact Abelian group andG^is its dual consisting of characters onG. It is proven that the representation is irreducible, which allows us to define a covariant positive operator-valued measure (covariant POVM) generated by orbits of projective unitary representations ofG. The quantum tomography associated with the representation is discussed. It is shown that the integration over such a covariant POVM defines a family of contractions which are multiples of unitary operators from the representation. Using this fact, it is proven that the measure is informationally complete. The obtained results are illustrated by optical tomography on groups and by a measure with a density that has a value in the set of coherent states.