There is a question in Artin asking when does an inverrtible 2x2 real matrix send a lattice to itself.
The issue I have is with this question not being specific as to what kind of answer is expected. Any lattice has a basis consisting of two elements and a matrix fixes it if it sends integer combinations of these elements to other integer combinations of them. Hence an answer could be "whenever the matrix is integer valued under the change of basis to the lattice basis". This seems pretty vague, though, so I'm wondering if there is anything else that can be said.
If the lattice is spanned by $a$ and $b$. A matrix $M$ in the basis $a,b$ is a bijection of the lattice iff $M\in M_2(\mathbb{Z})$ and of determinant equal to $1$ or $-1$. Since the inverse is unique $M^{-1}$ is linear and $det(M^{-1})=1/det(M)$ should be an integer. If the determinant is not as before the map is just injective.