Strictly characteristic not implies fully invariant

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

Proof
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.