[Wiki, as already in the comments]
For the easy negatives, consider the Cayley graph with respect to the generating set $S=G$. The graph obtained is then a complete graph. It can be embedded in $\mathbb{R}^{G} = \{ (a_g)_{g \in G} \mid a_g \in \mathbb{R} \}$ as follows: $g \mapsto \delta_{g\cdot}$ where $\delta_{gh} = 1$ if $h=g$ and $0$ else. The edges between two vertices all have length $\sqrt{2}$.
The permutation matrices are linear isometris of $\mathbb{R}^G$ and will permute the vertices as desired. So the symmetries of the Cayley graph contain $S_{|G|}$ which is strictly larger than $G$ as soon as $|G| \geq 2$.