  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 possibly if the geometric genus and the
irregularity satisfy $p_g(S)=q(S)\in\{1,2\}$. In the exceptional cases, the
image of the homomorphism $\mathrm{Aut}_s(S)\rightarrow
\mathrm{Aut}(\mathrm{CH}_0(S)_\mathrm{alb})$ has order at most 3.
  Our 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 the trivial action on
$\mathrm{CH}_0(S)_\mathrm{alb}$. As a by-product we obtain that if $S$ is an
elliptic K3 surface, then $\mathrm{Aut}_f(S)\cap \mathrm{Aut}_s(S)$ acts
trivially on $\mathrm{CH}_0(S)_\mathrm{alb}$.
