Is there a structure on $\mathbb{R}$ with a given cardinality of isomorphisms

Question

We have, for example
$|\mathrm{Aut} (\mathbb{R})|= |\mathbb{R}^\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R,+,0})|= |\mathbb{R}^\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R,<})|= |\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R},<,+,0)|= |\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R,<,+,\times,0,1})|= 1$

1. If $\kappa \leq |\mathbb{R}^\mathbb{R}|$ , is there a structure on $\mathbb{R}$ with automorphism group of cardinality $\kappa$ ?

2. Can you give explicit examples for the finite case, $\aleph_0$, $\aleph_1$ and $\aleph_2$.
   I'm especially interested in common structures, that occur naturally, but I'd also be interested in any other exotic examples you have.