Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$

Question

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$

$H$ is the set of fixed points on $A$

$A$ : set, $H \le S_A$, $F(H) = \{ a \in A : \sigma (a) = a, \forall \sigma \in H \}$

Let $\tau \in N_{S_A}(H), \sigma \in H$

$\tau[F(\sigma)] = [F(\tau\sigma\tau^{-1})] = [F(\sigma)]$ Is this valid? and from this can we get $\tau[F(H)] = [F(\tau H\tau^{-1})] = F(H)$ ?

Let $\omega \in A \backslash F(H)$

$\tau[F(\omega)] = [F(\tau\omega\tau^{-1})] = [F(\omega)]$ (This step seems incorrect to do)

$\tau[A \backslash F(H)] = \tau[A] \backslash \tau[F(H)] = A \backslash \tau [F(H)] = A \backslash [F(\tau H \tau^{-1})] = A \backslash F(H) $

This seems possibly correct but I am not convinced I did it right