Let $\psi \colon M \to N$ be $C^\infty$. A smooth vector field X along $\psi$ (that is, $X \in C^\infty(M,T(N))$ and $\pi \circ X = \psi$) has local $C^\infty$ extensions in $N$ if given $m \in M$ there exists a neighborhood $U$ of $m$ and a neighborhood $V$ of $\psi(m)$ such that $\psi(U) \subset V$ and there is a $C^\infty$ vector field $Y$ on $V$ such that $Y \circ \psi|U = X|U$.

Does a $C^\infty$ vector field $X$ along an immersion $\psi \colon M \to N$ always have local $C^\infty$ extensions in $N$?