Cocentral Split Abelian Hopf Algebra Extensions from Crossed Cocycles

We study cocentral split abelian Hopf algebra extensions over an algebraically closed field of characteristic zero. The kernel is $k^V$ and the quotient is $kΓ$, where $V$ is finite abelian and $Γ$ acts on $V$. For a fixed action, we describe these extensions by crossed families of normalized group 2-cocycles on $V$, modulo changes of homogeneous section. We give the obstruction to lifting cohomology classes to such cocycle data. Using the Schur multiplier of $V$, we rewrite this obstruction as a bicharacter lifting problem; it vanishes when $V$ has odd exponent. We then apply the theory to permutation modules and to arithmetic reductions of Coxeter modules, including explicit dihedral and rank-one affine examples.