Summary: After the classification of the finite-dimensional Nichols algebras of diagonal type [I. Heckenberger, Invent. Math. 164, No. 1, 175–188 (2006; Zbl 1174.17011); Adv. Math. 220, No. 1, 59–124 (2009; Zbl 1176.17011)], the determination of its defining relations [I. E. Angiono, J. Eur. Math. Soc. (JEMS) 17, No. 10, 2643–2671 (2015; Zbl 1343.16022); J. Reine Angew. Math. 683, 189–251 (2013; Zbl 1331.16023)], and the verification of the generation in degree-\(s1\) conjecture [Angiono, loc. cit.], there is still one step missing in the classification of complex finite-dimensional Hopf algebras with abelian group, without restrictions on the order of the latter: the computation of all deformations or liftings. A technique towards solving this question was developed in [the first author et al., J. Pure Appl. Algebra 218, No. 4, 684–703 (2014; Zbl 1297.16027)], built on cocycle deformations. In this paper, we elaborate further and present an explicit algorithm to compute liftings. In our main result we classify all liftings of finite-dimensional Nichols algebras of Cartan type \(A\), over a cosemisimple Hopf algebra \(H\). This extends [the first author and H.-J. Schneider, Math. Sci. Res. Inst. Publ. 43, 1–68 (2002; Zbl 1011.16025)], where it was assumed that the parameter is a root of unity of order \(>3\) and that \(H\) is a commmutative group algebra. When the parameter is a root of unity of order 2 or 3, new phenomena appear: the quantum Serre relations can be deformed; this allows in turn the power root vectors to be deformed to elements in lower terms of the coradical filtration, but not necessarily in the group algebra. These phenomena are already present in the calculation of the liftings in type \(A_2\) at a parameter of order 2 or 3 over an abelian group [M. Beattie et al., Isr. J. Math. 132, 1–28 (2002; Zbl 1054.16027); M. Helbig; Commun. Algebra 40, No. 9, 3317–3351 (2012; Zbl 1271.16034)], done by a different method using a computer program. As a byproduct of our calculations, we present new infinite families of finite-dimensional pointed Hopf algebras.