3-subnormal not implies finite-automorph-join-closed subnormal

Statement
A join of finitely many 3-subnormal subgroups that are all automorphs of each other need not be a fact about::subnormal subgroup. In fact, even a join of two 3-subnormal subgroups that are automorphs of each other need not be a subnormal subgroup.

Similar facts

 * Join of two 3-subnormal subgroups may be proper and contranormal
 * 4-subnormal not implies finite-conjugate-join-closed subnormal

Opposite facts

 * 2-subnormality is conjugate-join-closed
 * 2-subnormal implies join-transitively subnormal
 * 3-subnormal implies finite-conjugate-join-closed subnormal

Proof
The counterexample for this is the same as the counterexample in join of two 3-subnormal subgroups may be proper and contranormal.