= Solution
For a finite generating set $S$ of $G$, the <Cayley graph> $\operatorname{Cay}_S(G)$ has vertex set $G$ and an edge from $g$ to $gs$ for every $g\in G$ and $s\in S\cup S^{-1}$; one may retain orientations and labels, or forget them.
As unlabelled undirected graphs, the Cayley graph of the <infinite cyclic group> $\mathbb Z$ with generator $1$ and the Cayley graph of the <infinite dihedral group>
$$
D_\infty=C_2*C_2=\langle a,b\mid a^2=b^2=1\rangle
$$
with generators $a,b$ are both the two-way infinite line. The first group is an <abelian group> and the second is not, so they are not isomorphic. Thus the requested example is the <infinite line as a Cayley graph> of \b[$\mathbb Z$ and $D_\infty$].
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