Automorph-conjugacy is not finite-join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., automorph-conjugate subgroup) not satisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .
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Statement

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

Facts used

For the Hall subgroups example

1. Sylow implies automorph-conjugate
2. Hall implies join of Sylow subgroups
3. Hall not implies automorph-conjugate

Proof

The Hall subgroups example

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