= Solution
If two <order automorphisms> agree on $\mathbb Q$, their values at any real $x$ are the same <supremum> of the common values at rational points below $x$. This is the uniqueness argument for <extension of an order automorphism from a dense subset>; it does not require either automorphism to carry $\mathbb Q$ onto itself.
Their restrictions belong to $\mathbb R^{\mathbb Q}$, whose <cardinality> is
$$
(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}.
$$
This gives the upper bound. The translations $x\mapsto x+r$, one for each real $r$, are distinct <order automorphisms>, giving the lower bound. Consequently
$$
\boxed{|\operatorname{Aut}(\mathbb R,<)|=2^{\aleph_0}.}
$$
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