2 Explained rigidity in the question

In a recent question of mine I asked whether every infinite group is (isomorphic to) the automorphism group of a graph. The finite case was done by Frucht in 1939.

The first answer to this question pointed out two papers answering my original question, one by Sabidussi and one by de Groot.

Reading the 3-page paper by Sabidussi I thought "Wow, these graphs are huge": Sabidussi realizes a group of size $\kappa$ as the automorphism group of a graph of size $\aleph_\kappa$.

Indeed, de Groot in his paper notes that every countable group is the automorphism group of a countable graph, every group of size $\leq 2^{\aleph_0}$ is the automorphism group of a graph of size $\leq 2^{\aleph_0}$, and every group of size $\kappa$ is the the automorphism group of a graph of size $\leq 2^{\kappa}$.

But in general, he doesn't know how large a graph is needed to realize a given group.

Has this issue been resolved? Is there a reason why for a given infinite group $G$ there shouldn't be a graph of size $|G|$ whose automorphism group is isomorphic to $G$?