  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
$\mathfrak{sl}_2$, 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.
