  The aim of this note is twofold. Firstly, we explain in detail Remark 4.1 in
\cite{doan-a} by showing that the action of the automorphism group of the
second Hirzebruch surface $\mathbb{F}_2$ on itself extends to its formal
semi-universal deformation only up to the first order. Secondly, we show that
for reductive group actions, the locality of the extended actions on the
Kuranishi space constructed in \cite{doan-equivariant} is the best one could
expect in general.
