Liftable mapping class groups of certain branched covers of torus

Let Sg,n be a closed oriented hyperbolic surface of genus g with n marked points, with the understanding that Sg,0=Sg. Let Mod(Sh,n) be the mapping class group of Sh,n and LModₚ(Sh,n) be the liftable mapping class group associated to a cover p:Sg→ Sh,n. For the cover pₖ:Sₖ→ S1,2, Ghaswala, in his PhD thesis, derived a finite presentation for LModpₖ(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 LModpₖ(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 LModpₖ(S1,2) is maximal in Mod(S1,2) if and only if k is prime.