Liftable mapping class groups of certain branched covers of torus

Let S₆, ₍ be a closed oriented hyperbolic surface of genus g with n marked points, with the understanding that S₆, ₀=Sg. Let Mod (S₇, ₍) be the mapping class group of S₇, ₍ and LModₚ (S₇, ₍) be the liftable mapping class group associated to a cover p: Sg S₇, ₍. For the cover pₖ: Sₖ S₁, ₂, Ghaswala, in his PhD thesis, derived a finite presentation for LMod₏䂵 (S₁, ₂) when k=2, 3, 4 and a finite generating set when k=5, 6 using the Reidemeister-Schreier rewriting process. In this paper, we derive a finite generating set for LMod₏䂵 (S₁, ₂) for all k 2. In the process, we also prove that the kernel of the homology representation Ψ: Mod (S₁, ₂) GL₃ () is normally generated by a Dehn twist about a separating simple closed curve, and it is free with a countable basis. We also provide an explicit countable basis for Ψ consisting of separating Dehn twists. As an application of Birman-Hilden theory, we provide a finite generating set for the normalizer of the Deck group of pₖ in Mod (Sₖ) when k=2, 3. We conclude the paper by proving that LMod₏䂵 (S₁, ₂) is maximal in Mod (S₁, ₂) if and only if k is prime.