The Euler-Poincaré characteristic of a given poset X is defined as the alternating sum of the orders of the sets of chains Sd_n(X) with cardinality n + 1 over the natural numbers n. Given a finite gorup G, Thévenaz extended this definition to G-posets and defined the Lefschetz invariant of a G-poset X as the alternating sum of the G-sets of chains Sd_n(X) with cardinality n+1 over the natural numbers n which is an element of Burnside ring B(G). Let A be an abelian group. We will introduce the notions of A-monomial G-posets and A-monomial G-sets, and state some of their categorical properties. The category of A-monomial G-sets gives a new description of the A-monomial Burnside ring B_A(G). We will also introduce Lefschetz invariants of A-monomial G-posets, which are elements of B_A(G). An application of the Lefschetz invariants of A-monomial G-posets is the A-monomial tensor induction. Another application is a work in progress that aims to give a reformulation of the canonical induction formula for ordinary characters via A-monomial G-posets and their Lefschetz invariants. For this reformulation we will introduce A-monomial G-simplicial complexes and utilize the smooth G-manifolds and complex G-equivariant line bundles on them.