Extension of a group action beyond the boundary Let $M$ be a compact manifold with boundary and suppose a compact group $G$ acts on it. Can one always extend the action beyond the boundary? More precisely, does there always exist a $G$-manifold with boundary $N$, with $\dim M=\dim N$,  and a $G$-equivariant embedding of $M$ into the interior of $N$?
 A: If you are asking for a topological action, then it would seem you can just double $M$ along the boundary. I am not sure this works in the smooth category..
A: Yes, the action can be extended. It immediately follows from the equivariant collaring  theorem proven in the paper Kankaanrinta, Marja Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions. Algebr. Geom. Topol. 7 (2007), 1–27. 
Let me formulate the equivariant collaring theorem.  An equivariant collar of $\partial M$ is a smooth $G$-equivariant embedding
$$f:\partial{}M\times [0,\infty)\to M,$$ 
such that $f(x,0)=x$ for all $x\in \partial M$. The equivariant collaring theorem states that given a smooth proper action of a (possibly non-compact) Lie group an manifold $M$ with boundary $\partial{}M$, the boundary has an equivariant collar. 
To extend the action of $G$ "beyond the boundary" we now simply consider the manifold $M\cup_{\partial M}(\partial M\times(-1,0])$ and  extend the action of $G$ to $\partial M\times(-1,0]$ by
$$g\cdot (x,t)\ := \ (g\cdot x,t).$$
