  One may ask which maps between Hilbert modules allow for a completely
positive extension to a map acting block-wise between the associated (extended)
linking algebras. In these notes we investigate in particular those of such
CP-extendable maps whose 22-corner is a homomorphism, the CP-H-extendable maps.
We show that they coincide with the maps considered by Asadi [Asa09], by Bhat,
Ramesh, and Sumesh [BRS12], and by Skeide [Ske10]. We also give an intrinsic
characterization that generalizes the characterization by Abbaspour and Skeide
[AbSk07] of homomorphicly extendable maps as those which are ternary
homomorphisms. For general strictly CP-extendable maps we give a factorization
theorem that generalizes those of Asadi, of Bhat, Ramesh, and Sumesh, and of
Skeide for CP-H-extendable maps. As an application, we examine semigroups of
CP-H-extendable maps, so-called CPH-semigroups, and illustrate their relation
with a sort of generalized dilation of CP-semigroups, CPH-dilations.
