If H is a subgroup of an Abelian p-group G, we say G is H-fully transitive if using the height valuation from G, for every x ∈ H, every valuated (i.e., non-height decreasing) homomorphism x → G extends to a valuated homomorphism H → G. This notion is a generalization of the classical definition of fully transitive groups due to Kaplansky. A number of interesting properties of this idea are established.