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Abstract: In this talk, the representation rings (or the Green rings) for a family of Hopf algebras of tame type, the $2$-rank Taft algebra (at $q=-1$) and its two relatives twisted by $2$-cocycles are explicitly described via a representation theoretic analysis. It turns out that the Green rings can serve to detect effectively the twist-equivalent Hopf algebras here. This is a joint work with Dr. Yunnan Li.It is well-known that Drinfeld twist is a key method to yielding new Hopf algebras in quantum groups theory. Dually, 2-cocycle twist or Doi-Majid twist including Drinfeld double as a kind of such twist is extensively employed in various current researches. For instance, Andruskiewitsch et al considered the twists of Nichols algebras associated to racks and cocycles. Guillot-Kassel-Masuoka got some examples by twisting comodule algebras by 2-cocycles. In generic case, Pei-Hu-Rosso, Hu-Pei found explicit 2-cocycle deformation formulae between multi-(resp.two-)parameter quantum groups and one-parameter quantum groups Uq,q−1(g) and an equivalence between the weight module categories O as braided tensor ones. Likewise, in root of unity case, when a 2-cocycle twist exists under some conditions on the parameters, Benkart et al used a result of Majid-Oeckl to give a category equivalence between Yetter-Drinfeld modules for a finite-dimensional pointed Hopf algebra H and those for its cocycle twist Hσ, and further to derive an equivalence of the categories of modules for ur,s(sln) and uq,q−1 (sln) as Drinfeld doubles. In contrast, for particular choices of the parameters, there is no such cocycle twist, and in that situation the representation theories of ur,s(sln) and uq,q−1(sln) can be quite different. Recently, Bazlov-Berenstein considered cocycle twists and extensions of braided doubles in a broader setting including twisting the rational Cherednik algebra of the symmetric group into the Spin Cherednik algebra. A natural question is to ask how to detect two twist-equivalent Hopf algebras in nature? The article seeks to address this question through investigating the representation rings for a family of Hopf algebras, the 2-rank Taft Hopf algebra (at q = −1), and its two relatives twisted by 2-cocycles.