  We consider semigroup algorithmic problems in the Special Affine group
$\mathsf{SA}(2, \mathbb{Z}) = \mathbb{Z}^2 \rtimes \mathsf{SL}(2, \mathbb{Z})$,
which is the group of affine transformations of the lattice $\mathbb{Z}^2$ that
preserve orientation. Our paper focuses on two decision problems introduced by
Choffrut and Karhum\"{a}ki (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 $\mathsf{SA}(2, \mathbb{Z})$. We show that
both problems are decidable and NP-complete. Since $\mathsf{SL}(2, \mathbb{Z})
\leq \mathsf{SA}(2, \mathbb{Z}) \leq \mathsf{SL}(3, \mathbb{Z})$, our result
extends that of Bell, Hirvensalo and Potapov (2017) on the NP-completeness of
both problems in $\mathsf{SL}(2, \mathbb{Z})$, and contributes a first step
towards the open problems in $\mathsf{SL}(3, \mathbb{Z})$.
