Rademacher Complexity of the Composition with an Indicator
In general, composing a threshold function with a function class can arbitrarily increase the Rademacher complexity (so the answer to your question is negative). Let $\Omega$ be a finite set, $|\Omega|=N$, and let $F=[-a,a]^\Omega$ be the set of all functions from $\Omega$ to $[-a,a]$.
When $N\gg n$, the Rademacher complexity (either empirical or expected) of $F$ will be $\Theta(a)$. Let $F'$ be $F$ composed with the sign function; now $F'=\{-1,1\}^\Omega$ and its Rademacher complexity in this regime is $\Theta(1)$. Since $a&gt;0$ can be taken arbitrarily small, this shows that the multiplicative gap can be arbitrarily large.