Hopfian combinatorial wreath products

Let $A$ be an abelian group. We consider sufficient conditions for the combinatorial wreath product $A \wr_X B$ to be Hopfian generalising results of Bradford and Fournier-Facio. For an integer $m \geq 2$ we show an example where $\mathbb{Z}/ \mathbb{Z}_m \wr_X B$ is not Hopfian but $B$ is Hopfian. We describe $Aut(A \wr_X B)$ under some restrictions on $A$, $B$ and $X$.