Cayley and character tables for the lamplighter group

Let us define the lamplighter group $$L(G)$$ on the group $$G$$ is defined as a semi direct product, $$L(G) := \mathbb{G} \ltimes \sum_{x\in G}\mathbb{Z}_2$$, with the direct sum of copies of $$\mathbb{Z}_2$$ indexed by $$G$$; for $$m,m' \in G$$ and $$\eta,\eta' \in \sum_{x\in \mathbb{Z}}\mathbb{Z}_2$$ the group operation is:

$$(m,\eta)(m', \eta') := (m + m',\eta \oplus \rho^{-m}\eta')$$ where $$\oplus$$ is component wise addition modulo 2 and is right shift.

Let us consider a simple case taking $$G$$ as $$\mathbb{Z}_2$$ x $$\mathbb{Z}_2$$.

how would I construct the Cayley table and characters for this particular group?

EDIT: So I did some trivial calculations and noticed the Cayley table is similar to the Klein-$$4$$ group. However we know $$\rho$$ sends $$0 \rightarrow 1$$ and the inverse is vice-versa. Now there are some 'issues' with four entires of the table where $$m$$ takes on the value of $$1$$.

Using the group definition, the Cayley table reads:

\begin{align*} \begin{array}{c | c c c c } & (0,0) & (0,1) & (1,0) & (1,1)\\ \hline (0,0) & (0,0) & (0,1) & (1,0) & (1,1)\\ (0,1) & (0,1) & (0,0) & (1,1) & (1,0)\\ (1,0) & (1,0) & ? & (0,0) & ?\\ (1,1) & (1,1) & ? & (0,1) & ?\\ \end{array} \end{align*}

• You need to provide more context or other details, and your question (whether interesting or not!) is likely to be closed if you do not edit your question to provide additional context. This context should ideally explain why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc. – user1729 Feb 20 at 11:58
• Well I'm doing research and I asked myself this question so I wanted to find a solution. – Math Feb 20 at 12:00
• Sure, but then why can't you answer it yourself? Where did you get stuck? Saying these things count as "context". – user1729 Feb 20 at 12:02
• Please look at my edited question. – Math Feb 20 at 12:05
• What's $\Bbb G$? Why is one sum indexed over $x\in G$ but the other over $x\in\Bbb Z$? What is right shift? – Christoph Feb 20 at 12:08