Automorph-conjugacy is not finite-join-closed

Statement
A join of finitely many automorph-conjugate subgroups need not be automorph-conjugate.

For the Hall subgroups example

 * 1) uses::Sylow implies automorph-conjugate
 * 2) uses::Hall implies join of Sylow subgroups
 * 3) uses::Hall not implies automorph-conjugate

The Hall subgroups example
The proof follows directly from facts (1)-(3).