  We introduce and study the notion of the $G$-Tutte polynomial for a list
$\mathcal{A}$ of elements in a finitely generated abelian group $\Gamma$ and an
abelian group $G$, which is defined by counting the number of homomorphisms
from associated finite abelian groups to $G$.
  The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte
polynomial for realizable (arithmetic) matroids, the characteristic
quasi-polynomial for integral arrangements, Br\"and\'en-Moci's arithmetic
version of the partition function of an abelian group-valued Potts model, and
the modified Tutte-Krushkal-Renhardy polynomial for a finite CW-complex.
  As in the classical case, $G$-Tutte polynomials carry topological and
enumerative information (e.g., the Euler characteristic, point counting and the
Poincar\'e polynomial) of abelian Lie group arrangements.
  We also discuss differences between the arithmetic Tutte and the $G$-Tutte
polynomials related to the axioms for arithmetic matroids and the
(non-)positivity of coefficients.
