Are fixed points of complements have same size?

Question

Let $G$ be a finite group. $G=NA=NB$ with $N\cap A=N\cap B=1$ and $N$ is normal in $G$.

Can we say that $$|C_N(A)|=|C_N(B)|$$ ?

If complements $A,B$ are conjugates in $G$, of course we can say this. Can we say this in general ?

If it is true, Is $C_N(A)$ isomorphic to $C_N(B)$ ?

Answer 1

No. For example $A_7$ has complements $\langle (1,2) \rangle$ and $\langle (1,2)(3,4)(5,6) \rangle$ in $S_7$, and their centralizers (in $S_7$) have orders $240$ and $48$, respectively