Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$. 

1. Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still conjugate Weil numbers and by Honda-Serre-Tate should correspond to an abelian variety $\tilde A/\mathbb F_{q^d}$ of the same dimension as $A$. What is the relation between $\tilde A$ and $A$?

2. Consider the $nd$ numbers $\{q^{1/2}\alpha_i^{1/d}\}$. The Abelian variety $\tilde A'$ corresponding to these numbers is, I believe the Weil Restriction of $\tilde A$ from $\mathbb F_{q^d}$ to $\mathbb F_q$. Is there a direct relation between $\tilde A'$ and $A$?  

(Of course, Honda-Tate is only upto isogeny but I would be very happy if there was a functor upto isomorphism that picked out some canonical $\tilde A,\tilde A'$.)