Wilfrid Hodges has shown that it is consistent with ZF that there be is an algebraic closure $L$ of the rational field $\mathbb{Q}$ with no nontrivial automorphisms. Obviously $|Aut(L)\smallsetminus {1}| = 2^{\aleph_{0}}$.
See: W. Hodges, Läuchli's algebraic closure of $\mathbb{Q}$. Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 2, 289--297
Wilfrid Hodges has shown that it is consistent with ZF that there be an algebraic closure $L$ of the rational field $\mathbb{Q}$ with no nontrivial automorphisms. Obviously $|Aut(L)\smallsetminus {1}| = 2^{\aleph_{0}}$.
See: W. Hodges, Läuchli's algebraic closure of $\mathbb{Q}$. Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 2, 289--297