Suppose $M$ is a topological manifold and $\iota: N\hookrightarrow M$ be a submanifold. A normal microbundle of $N$ consists of an open neighborhood $U$ of $N$ and a retraction $\pi: U \to N$ such that $(U, N, \iota, \pi)$ is a microbundle. It is known that if a normal microbundle exists, then it is unique up to isomorphism after certain stabilization, i.e., unique when we consider instead the induced embedding $N \hookrightarrow M \times \{0\}\hookrightarrow M \times {\mathbb R}^k$. However, without doing stabilization we do not have uniqueness.

On the other hand, in the smooth category, we know that tubular neighborhoods of a smooth submanifold $N \subset M$ exists and is unique upto isotopy. There is no need to do stabilization. By a tubular neighborhood I mean an open neighborhood $U$ of $N$ together with a smooth retraction $\pi: U \to N$ which restricts to the identity on $N$. Of course, a tubular neighhborhood induces a normal microbundle.

Question A: suppose $M$ is a smooth manifold and $N \subset M$ is a smooth submanifold. Are all normal microbundles isotopic, hence isotopic to the one coming from the smooth structure? Isotopy means the following: in this case a normal microbundle must be homeomorphic to a disk bundle inside the normal bundle of the smooth embedding, or equivalently a continuous open embedding of a disk bundle of the normal bundle to a neighborhood of $N$. Two normal microbundles are isotopic if the corresponding open embeddings are isotopic via open embeddings fixing $N$.

Question B: More generally, suppose $M$ is acted smoothly by a finite group $G$ and $N$ is $G$-invariant, are all $G$-equivariant normal microbundles $G$-isotopic, i.e., isotopic via $G$-equivariant open embeddings which fix $N$? The typical situation which I care about is when $N$ is the fixed point set of the $G$-action.