  In this note we study the automorphism group of a smooth Danielewski surface
$D_p= \{(x,y,z) \in \mathbb{A}^3 \mid xy = p(z) \} \subset \mathbb{A}^3$, where
$p \in \mathbb{C}[z]$ is a polynomial without multiple roots and $deg (p) \ge
3$. It is known that two such generic surfaces $D_p$ and $D_q$ have isomorphic
automorphism groups. Moreover, $\mathrm{Aut}(D_p)$ is generated by algebraic
subgroups and there is a natural isomorphism $\phi \colon \mathrm{Aut}(D_p)
\xrightarrow{\sim} \mathrm{Aut}(D_q)$ which restricts to an isomorphism of
algebraic groups $G \xrightarrow{\sim} \phi(G)$ for any algebraic subgroup $G
\subset \mathrm{Aut}(D_p)$. In contrast, we prove that $\mathrm{Aut}(D_p)$ and
$\mathrm{Aut}(D_q)$ are isomorphic as ind-groups if and only if $D_p \cong D_q$
as a variety. Moreover, we show that any automorphism of the ind-group
$\mathrm{Aut}(D_p)$ is inner.
