We consider semigroup algorithmic problems in the Special Affine group SA(2,Z)=Z2⋊SL(2,Z), which is the group of affine transformations of the lattice Z2 that preserve orientation. Our paper focuses on two decision problems introduced by Choffrut and Karhumäki (2005): the Identity Problem (does a semigroup contain a neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of SA(2,Z). We show that both problems are decidable and NP-complete. Since SL(2,Z)≤SA(2,Z)≤SL(3,Z), our result extends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completeness of both problems in SL(2,Z), and contributes a first step towards the open problems in SL(3,Z).