Second invariant cohomology of some finite-dimensional Hopf algebras

We use categorical description of the invariant 2-cohomology group of Hopf algebra to compute such cohomology for two finite dimensional Hopf algebras: the group ring of $Z_8⋊Aut(Z_8)$ and Kac-Paljutkin algebra. For the first of these two examples, our categorical approach helps to settle the problem of computing this cohomology, which was left open in by Guillot and Kassel (\cite{dtwist}), where only some partial information about this cohomology was obtained.