Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$f(n)=n-2^{\ell(n)}$$ Let $a(n)$ be a permutation of the nonnegative integers such that $a(0)=0$, $a(n)=n$ if $n$ is a power of $2$ and subsequences from $a(2^m)$ to $a(2^{m+1}-1)$ for $m\geqslant0$ contain all number $k$ such that $2^m\leqslant k < 2^{m+1}$.

Let $b(n)$ be a sequence of the nonnegative integers such that $b(0)=0$, $b(n)=n$ if $n$ is a power of $2$, otherwise $$b(n)=a(n)+b(f(n))-f(n)$$

Is there at least one permutation of the nonnegative integers $a(n)$ such that $b(n)$ is also a permutation of the nonnegative integers?

This question is motivated by my previous question, where the Fibonacci analogs of $\ell(n),f(n),a(n)$ and $b(n)$ has the given property.