Let $M$ be a normal extension of $F$. Suppose that $a_1, a_2$ are in $M$ and are the roots of the minimal polynomial of $a_1$ over $F$, and $b_1,b_2$ are the roots of minimal polynomial of $b_1$ over $F$. Determine whether or not there is an $F$-automorphism $Z$ from $M$ to $M$ with $Z(a_1)=a_2$, and $Z(b_1)=b_2$.
As Dylan points out in the comments, it is not clear whether you want $a_1,a_2$ to be the only roots of the minimal polynomial of $a_1$; likewise, $b_1,b_2$. I'll point you toward a counterexample in the case where they are not the only roots. Let the first polynomial have distinct roots $a_1,a_2,a_3$ (and perhaps others), let $b_i=2a_i$ for all $i$; you won't find an automorphism that takes $a_1$ to $a_2$ and $b_1$ to $b_3$.
EDIT: in partial and elliptical response to awllower's comment, there's no automorphism taking $\sqrt2$ to $-\sqrt2$ and also taking $\root4\of2$ to $-\root4\of2$.