I guess this could be a hard question but since I am no expert, and I am not even close to an expert I really do not know.
The basic idea is that we start with some group $(G,*)$. Because $(G,*)$ is a group it has over itself defined operation $*$ which satisfies all the group axioms.
Now, suppose that we take the set $G$ and that we seek to find another operation $o$ such that $(G,o)$ is also a group.
It seems to me that for some sets we will be able to find another operation under which the set is a group and for some sets we will not (but I may be mistaken).
So the question would be:
Are there any necessary, or sufficient, or necessary and sufficient conditions on the set $G$, or on the set $G$ and operation $*$, such that under these conditions there exist (or do not exist) at least one more operation (call it $o$) such that $(G,o)$ is also a group? What is known on these matters?