On generalizations of Lavrentieff's theorem for Polish group actions

It is shown that for every Polish group $G$ that is not locally compact there is a continuous action $a$ of $G$ on a $\boldsymbol{\Pi}^1_1$-complete subset $A$ of a Polish space $X$ such that $a$ cannot be extended to any superset of $A$ in $X$. This answers a question posed by Becker and Kechris and shows that an earlier theorem of them is optimal in several aspects.