Index two not implies characteristic

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup of index two) need not satisfy the second subgroup property (i.e., characteristic subgroup)
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Statement

A subgroup of index two in a group need not be a characteristic subgroup.

Proof

Example of the Klein four-group

Further information: Klein four-group