I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references.

The lamplighter group is defined by the following presentation: $$ L_N=(\mathbb{Z} _N) \wr \mathbb{Z} \cong\left\langle t, a_n, n \in \mathbb{Z} \mid a_n^N, t a_n t^{-1}=a_{n+1}, n \in \mathbb{Z}, a_n a_m a_n^{-1} a_m^{-1}, m, n \in \mathbb{Z}\right\rangle . $$

In this paper about AUTOMORPHISMS OF HIGHER RANK LAMPLIGHTER GROUPS, they computed the automorphism group of a large class of groups containing the Lamplighter group.

Theorem 3.2. If $d \geq 2$, then $$ \operatorname{Aut}\left(\Gamma_d(q)\right) \cong \operatorname{Der}\left(\mathbb{Z}^{d-1}, \mathcal{R}_d\left(\mathcal{L}_q\right)\right) \rtimes\left(U\left(\mathcal{R}_d\left(\mathbb{Z}_q\right)\right) \rtimes \mathcal{K}\right) $$ where $\mathcal{K}=\left\{\beta \in \operatorname{Aut}\left(\mathbb{Z}^{d-1}\right) \mid K^\beta=K\right\}$.

(When $d=2$, we have the lamplighter groups)

I was wondering if there is a more straightforward description & computation specifically for the automorphism of the lamplighter groups.

Any reference for this would be really appreciated.