Starting from an abelian group G and a factorizable ribbon Hopf G-bialgebra H, we construct a TQFT J H for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in G. When restricted to the subcategory of cobordisms with trivial decorations, our functor recovers a special case of Kerler-Lyubashenko TQFTs, namely those associated with factorizable ribbon Hopf algebras. Our result is inspired by the work of Blanchet-Costantino-Geer-Patureau, who constructed non-semisimple TQFTs for admissible decorated cobordisms using the unrolled quantum group of sl 2 , and by that of Geer-Ha-Patureau, who reformulated the underlying invariants of admissible decorated 3-manifolds using ribbon Hopf G-coalgebras. Our work represents the first step towards a homological model for non-semisimple TQFTs decorated with cohomology classes that appears in a conjecture by the first two authors.