Extending smooth functions on a submanifold to G-invariant functions

Question

Let $M$ be a smooth manifold and $N\subseteq M$ a submanifold (say embedded). Let $G$ be a Lie group acting on $M$ and let $H$ be a closed Lie subgroup of $G$ which leaves $N$ invariant and such that if $g\in G$ is such that $g\cdot p\in N$ for some $p\in N$, then $g\in H$. That is, $G$ acts on $N$ only through $H$. Let $f\in C^\infty(N)$ be an $H$-invariant function. Can it always be extended to a $G$-invariant smooth function on $M$ (at least locally)?

Answer 1

This is not the case. Consider the action of $\mathbb{R}$ on $\mathbb{R}^2$ given by $$ r\cdot(x,y)=(x+ry,y) $$ This action preserves all horizontal lines, acting trivially on the $x$-axis and transitively on the rest. Letting $G=H=\mathbb{R}$, $M=\mathbb{R}^2$, and letting $N$ be the $x$-axis in $\mathbb{R}^2$, we have that any smooth function $N\to\mathbb{R}$ is preserved by $H$, but all smooth $G$-invariant functions $M\to\mathbb{R}$ are constant w.r.t. $x$.