Terminology: The group(s) of symmetries of the Cayley graph of a group.

Please forgive me if this question is ill-defined. It's late here and I want to ask the question whilst it's still fresh in my mind.

Motivation:

Suppose we have a group $G$ given by a presentation $$P_G=\langle a, b\mid a^2, b^3, (ab)^2\rangle.$$ Then $P_G$ has the (oriented) graph $\Gamma_G$ in Figure 1: (This image can be found on page 58 of Magnus et al.'s Combinatorial Group Theory [ . . . ].)

The group $H_{\Gamma_G}$ of symmetries of $\Gamma_G$ as it is is $\Bbb Z_3$. If we were to extend the graph into the three dimensional world to create a prism-like shape $\Pi_G$, then the group $H_{\Pi_G}$ of symmetries of $\Pi_G$ would be $\Bbb Z_3\times\Bbb Z_2$.

So here is my motivating question:

What are $H_{\Gamma_G}$ and $H_{\Pi_G}$ called with respect to $G$?

I'm more interested $H_{\Pi_G}$ because it seems more "optimal". (I hope you can see why.)

What I mean by "with respect to $G$" is something like "the Shaun group(s) of $G$", if you'll forgive me using my name as a hypothetical example.

The Question:

What is the terminology for the group or groups of symmetries of the Cayley graph(s) of a group $G$?

Context:

I'm not sure how to provide additional context for a terminology question but here goes . . .

I'm studying for a PhD in combinatorial group theory and I'm in the first year.

I haven't found anything ibid. nor in Lyndon & Schupp's book "Combinatorial Group Theory".

I have no training in graph theory.

An answer that defines the group(s) in question and provides examples of their use in group theory literature would be ideal and greatly appreciated.

I'm just curious.

• I found this; there's no clear terminology for the groups though. – Shaun Aug 5 '18 at 1:37
• Related, but not a duplicate. – Shaun Aug 5 '18 at 1:39
• It seems like you are finding the group of symmetries, not of your graph, but of the way the graph is drawn - which is not well-defined. The graph in your first picture already has the symmetry group $\mathbb Z_2 \times \mathbb Z_3$. (Or possibly $D_3$? The Cayley graph you drew doesn't actually seem to match the group presentation, since $ab$ and $b^2$ wouldn't be connected by an $a$ edge if $a$ and $b$ commute.) – Misha Lavrov Aug 7 '18 at 17:32
• I mean the symmetries of the graph, @MishaLavrov. I thought that was clear from how I emphasised $H_{\Pi_G}$. – Shaun Aug 7 '18 at 17:36
• Sorry, $P_G=\langle a, b\mid a^2, b^3, (ab)^2\rangle.$ – Shaun Aug 7 '18 at 17:39

It's worth pointing out that the automorphism group you get depends on the generators you use to draw the Cayley graph, not just the group itself. For example, rather trivially, you can take a group $G$ and draw the Cayley graph with respect to the generating set $G$ (that is, take all elements of the group as generators; if loops upset you, leave out the identity element). This will be a complete directed graph on $|G|$ vertices, so its automorphism group will be the symmetry group of order $|G|$.
From the other end, we know that $G$ itself is always a subgroup of the automorphism group of any Cayley graph of $G$: for any $g \in G$, left multiplication by $g$ is an automorphism of any Cayley graph.
There is some interest in finding generating sets such that the automorphism group of the Cayley graph is exactly $G$, but this is not always possible. (See the article I linked to above for more details).