Let \((X,+)\) be a group and \((Y,+)\) be a commutative monoid. We prove that the Cauchy nucleus of a set-valued map \(F:X\rightarrow 2^Y\setminus \{\emptyset \},\) i.e. the set \(\{y\in X:\,F(x+y)=_KF(x)+F(y)\;\text{ for } \text{ every }\;x\in X\},\) is a subgroup of X provided it is nonempty, and that every subgroup of X is the Cauchy nucleus of a set-valued map \(F:X\rightarrow 2^Y\setminus \{\emptyset \}\) . We apply these results to characterize solutions of a partially Pexiderized Cauchy equation \(F(x+y)=_KF(x)+G(y)\) for every \((x,y)\in X\times S\) , and also solutions of this equation satisfied almost everywhere (in the sense of an ideal) in \(X\times S\) , provided S is a subgroup of X.
