  Let $R_n$ denote the KLR algebra of type $A^{(1)}_{e-1}$. Using the
presentation of Specht modules given by Kleschev-Mathas-Ram, Loubert completely
determined $\hom_{R_n}(S^\mu,S^\lambda)$ where $\mu$ is an arbitrary partition,
$\lambda$ is a hook and $e\neq2$. In this paper, we investigate the same
problem when $e=2$. First we give a complete description of the action of the
generators on the basis elements of $S^\lambda$. We use this result to identify
a large family of partitions $\mu$ such that there exists at least one non-zero
homomorphism from $S^\mu$ to $S^\lambda$, explicitly describe these maps and
give their grading. Finally, we generalise James's result for the trivial
module.
