Let's say you have a group like $(\mathbb{R}, +)$, where it’s not immediately clear where the symmetries are (compared to a group $(S, \circ)$ which is a group of symmetries of some object equipped with composition).
I'm trying to come up with a procedure to turn any group into another group where the symmetries are more obvious.
On a high level, I can use Cayley's theorem to find a subgroup of a symmetric group that is isomorphic to my original group.
I want to know what happens on a low level. My current hypothesis is that you consider the symmetries of the group itself, put them in a set, and equip these transformations with composition, you get the group I'm looking for. I presume what I have described is Cayley's theorem itself.
This checks out for the group $(\mathbb{R}, +)$, where the symmetries of the group are the translations on $\mathbb{R}$.
Is what I'm saying correct?