  Let $S$ be a complex smooth projective surface of Kodaira dimension one. We
show that the group $\mathrm{Aut}_s(S)$ of symplectic automorphisms acts
trivially on the Albanese kernel $\mathrm{CH}_0(S)_{\mathrm{alb}}$ of the
$0$-th Chow group $\mathrm{CH}_0(S)$, unless the geometric genus and the
irregularity satisfy $p_g(S)=q(S)\in\{1,2\}$. In the exceptional case, the
image of the homomorphism $\mathrm{Aut}_s(S)\rightarrow
\mathrm{Aut}(\mathrm{CH}_0(S)_{\mathrm{alb}})$ is either trivial or possibly
isomorphic to $\mathbb{Z}/3\mathbb{Z}$.
  Our main arguments actually take care of the group $\mathrm{Aut}_f(S)$ of
fibration preserving automorphisms of elliptic surfaces $f\colon S\rightarrow
B$. We prove that, if $\sigma\in\mathrm{Aut}_f(S)$ induces the trivial action
on $H^{i,0}(S)$ for $i>0$, then it induces trivial action on
$\mathrm{CH}_0(S)_\mathrm{alb}$. Also, if $S$ is additionally a K3 surface,
then $\mathrm{Aut}_f(S)\cap \mathrm{Aut}_s(S)$ acts trivially on
$\mathrm{CH}_0(S)_{\mathrm{alb}}$.
