Characteristic not implies quasiautomorphism-invariant

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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., characteristic subgroup) need not satisfy the second subgroup property (i.e., quasiautomorphism-invariant subgroup)
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It is possible to have a group G and a characteristic subgroup H of G such that H is not a quasiautomorphism-invariant subgroup, i.e., there are quasiautomorphisms of G that do not send H to itself.

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Further information: inner automorphism group of wreath product of groups of order p

Suppose p is a prime number greater than 3. Let G be the group isomorphic to the inner automorphism group of the wreath product of two groups of order p. G is a group of order p^p with an elementary abelian normal subgroup N of order p^{p-1}, an element of order p acting on it from outside, and every non-identity element of G has order p.

Consider the subgroup H = [G,G]. H is a group of order p^{p-2}, contained inside the elementary abelian normal subgroup. It is a characteristic subgroup.

However, we can construct a quasiautomorphism \sigma of G that does not preserve H as follows: the restriction of \sigma to N is an automorphism of N that fixes the center Z(G) (which is cyclic of order p) but does not send H to itself, and \sigma fixes every element of G outside H. Note that we need p > 3 to ensure that H is strictly bigger than Z(G), which is necessary to be able to construct a \sigma with the desired specifications.