Let $M,N$ be two closed differential manifolds and let $G$ be a compact Lie group. Assume that $G$ acts on both manifolds $M,N$ nicely (i.e. free and proper so that $M/G$ and $N/G$ have the structure of manifolds, but I don't think that this is important for my question). Furthermore, assume that there exists open neighborhoods $U_1\subset M$ and $V_1 \subset N$ and a map $$s_1:U_1\rightarrow V_1.$$

**Question:** Is it possible to construct $G$-invariant open subset $U\subset M$ and $V\subset N$ and a map $s:U\rightarrow V$ such that $\forall g\in G$ and $\forall p \in U$ we have $$s(g\bullet_{M} p)=g\bullet_{N}s(p),$$ where $\bullet_{M}$ and $\bullet_{N}$ denote the $G$-action in both $M$ and $N$, respectively and such that $U_{1} \subset U$, $V_{1} \subset V$ and there exists some subset $A\subset U_{1}$ such that $s|_{A} \equiv s_{1}$. If this is in general not possible, then under what conditions on $s_1$ can this be realized?

For an answer I would be thankful!