Schur-triviality is not characteristic subgroup-closed

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This article gives the statement, and possibly proof, of a group property (i.e., Schur-trivial group) not satisfying a group metaproperty (i.e., characteristic subgroup-closed group property).
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Statement

It is possible to have a Schur-trivial group G and a characteristic subgroup H of G such that H is not a Schur-trivial group.

Proof

Example of semidihedral group

We take the following: