Forcing linearity numbers for abelian groups

Forcing linearity number of an abelian group A is defined as the infimum of cardinalities of sets S of proper subgroups of A such that any homogeneous map f : A --> A is an endomorphism of A whenever it is linear on each member of S. The forcing linearity number is determined for all abelian groups. These numbers can be 0, 1, 2, p + 2 (for primes p), or N-0. There is a pathological case: the direct sums of two non-zero cyclic p-groups, where such a number does not exist.