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 $\tilde A/\mathbb F_{q^d}$. 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 $A$?

In my particular case, I also have a $\mathbb Z[\zeta_{\ell^m}]$ action on $A$ which has dimension $\phi(\ell^m)$ for $\ell $ a a prime different from $q$, $m$ very large, $d = \ell$ and $\dim \tilde A = \ell\dim \tilde A'$. I don't know if this is relevant at all.

(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 $\tilde A,\tilde A'$.)