I've been contemplating various conjectures on various fields that a priori don't have anything to do with group theory; and yet these heuristics (that it would take too long to go over, and are besides the point here) seem to imply the following very weird proposition:

Fix a prime $p\geq 3$. Let $G$ be a finite, prime-to-$p$, group generated by $r$ elements: $a_1,...,a_r$. Assume $r\leq p$. Then there exists an automorphism $\phi \in Aut(G)$ such that $G/$the group normally generated by $a_i \phi(a_i)^{-1}$ (for every $i$) is generated by elements whose order divides $p-1$.

This is true for $r=1$ (pick $\phi(a_1)=a_1^{-1}$).

I can prove it for some easy examples. Can you think of a reason for this to be true/of a counterexample?