16.29 (2006)

This property holds when $A$ is a cyclic subgroup (J. Siemons, A. Zalesskii, J. Algebra, 256 (2002), 611–625), as well as when $A$ is contained in some maximal torus and $G = \text{PSL}_n(q)$ (J. Siemons, A. Zalesskii, J. Algebra, 226 (2000), 451–478). Note that if $G = L_2(5)$, $A = 2 \times 2$, and $H = 5:2$ (in Atlas notation), then $A^x \cap H > 1$ for every $x \in G$ (this example was communicated to the author by V. I. Zenkov).