Let A be an abelian group, not necessarily finite. The main objective of this paper is to provide two constructions for an A-fibered biset functor. The first is the lower plus construction, and the other is the upper plus construction. These constructions coincide with the lower plus and upper plus constructions for biset functors (see [2]) when the fiber is the trivial group \(\lbrace \cdot \rbrace \) . To generate these constructions, it is necessary to have a family \(\mathcal {G}\) of finite groups and a function \(\mathcal {S}\) such that, for all \(G, H \in \mathcal {G}\) , we relate them to a subset \(\mathcal {S}(G, H)\) of \(\mathcal {M}^A (G, H)= \mathcal {M}(G,H):=\lbrace (L,\varphi ) \mid L\le G\times H, \ \varphi \in \operatorname {Hom}(L,A)\rbrace \) The pair \(\mathcal {(G,S)}\) satisfies certain axioms outlined in Definition 3.1. These axioms are essential for enabling the pair \(\mathcal {(G, S)}\) to generate the subcategory \(\mathcal {D}\) of the category of A-fibered bisets, denoted as \(\mathcal {C}^A.\) . In Sect. 2, we recall and extend the basic definitions and results for fibered bisets and fibered biset functors. In Sect. 3, the upper and lower constructions are defined. For this purpose, the pair \(\mathcal {(G, S)}\) is associated with \(\mathcal {(G, S_+ )}\) and \(\mathcal {(G, S^+ )}\) , with which we can construct the subcategories \(\mathcal {D}_+\) and \(\mathcal {D}^+\) of \(\mathcal {C}^A\) . In Sects. 4 and 5, the constructions \(F_+\) and \(F^+\) are described for an A-fibered biset functor F over \(\mathcal {D}\) , which are fibered biset functors over \(\mathcal {D_+}\) and \(\mathcal {D^+}\) respectively. In Sect. 6, the mark morphism \(F_+ \longrightarrow F^+\) is defined, which is a natural transformation. Necessary conditions on the ring R are also provided for the mark morphism to be injective or even bijective. In Sect. 7 the additional condition that F has a multiplicative structure is added, more precisely, that F is a fibered Green functor over \(\mathcal {D}\) . Then it can be shown that \(F_+\) and \(F^+\) inherit the structure of fibered Green functors over \(\mathcal {D_+}\) and \(\mathcal {D^+ }\) , respectively, and the mark morphism is multiplicative. In Sect. 8, the ring of characters is defined as a fibered biset functor, and its construction \(-_+\) , which is the global functor of fibered representations, is calculated. Finally, in Sect. 9, it is proved that the functor related to the lower construction, denoted by \(-_+\) , is the left adjoint for a restriction functor.
