Let $X$ be a smooth quasi-projective irreducible variety over the field of complex numbers $\mathbb{C}$. We denote by $\mathrm{Aut}(X)$ the group of algebraic automorphisms of $X$. Moreover, for a variety $V$, we call a map $V \to \mathrm{Aut}(X)$ a *morphism*, if the induced map $V \times X \to X$
is a morphism of varieties. The connected component of $\mathrm{Aut}(X)$
of the neutral element $e \in \mathrm{Aut}(X)$ we define by
$$
\mathrm{Aut}(X)^\circ = \left\{ g \in \mathrm{Aut}(X) \Big|
\begin{array}{l}
\, \textrm{$\exists$
an irreducible variety $V$ and a morphism} \\
\textrm{$V \to \mathrm{Aut}(X)$ s.t. the image contains $g$ and $e$}
\end{array} \right\} \, .
$$
This notion goes back to Ramanujam, see [Ram64]. Clearly, $\mathrm{Aut}(X)^\circ$ is a normal subgroup of $\mathrm{Aut}(X)$.

I am interested in the size of the group $Q(X) = \mathrm{Aut}(X) / \mathrm{Aut}(X)^\circ$. In case $X$ is projective, then $Q(X)$ is countable. Also in case $X$ is affine, $Q(X)$ is countable. My question is, whether this is true in general, i.e. whether for all smooth irreducible quasi-projective variety $X$, the group $Q(X)$ is countable. Every proof, counter-example or textbook reference would be perfect.

[Ram64] *Ramanujam, C.P.*, **A note on automorphism groups of algebraic varieties**, Math. Ann. 156, 25-33 (1964). ZBL0121.16103.