The Tietze Extension Theorem states that if $V$ is a closed subset of a metric space $X$ and $f$ is a continuous function on $V$ then there exists $F$ continuous on $X$ such that $F|_{V} = f$.
Suppose that $G$ is a countable discrete group acting continuously on $X$ and that $f$ is $G$-invariant. Does there exist $F$ continuous on all of $X$ which is also $G$-invariant such that $F|_{V} = f$?
The key step to prove it seems to be a $G$-invariant Urysohn Lemma: if $U$ and $V$ are closed subsets of $X$ which are $G$-invariant, does there exist a $G$-invariant continuous function such that $f^{-1}(U) = \{ 0 \}$ and $f^{-1}(V) = \{ 1 \}$?