Strictly characteristic not implies fully invariant

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., strictly characteristic subgroup) need not satisfy the second subgroup property (i.e., fully invariant subgroup)
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Statement

A strictly characteristic subgroup of a group need not be a fully invariant subgroup.

Proof

Further information: center not is fully invariant

The center of a group is always a strictly characteristic subgroup. Thus, to show that a strictly characteristic subgroup need not be fully characteristic, it suffices to construct an example where the center is not fully characteristic. Such an example is found here.

More generally, any characteristic subgroup of finite group that is not fully invariant can work. Thus, the finite examples for characteristic not implies fully invariant all work.