Characteristic not implies fully invariant in class three maximal class p-group

Statement
Let $$p$$ be any prime number. There exists a $$p$$-group that is of class three that is a fact about::maximal class group with a fact about::characteristic subgroup that is not fully invariant.

Similar facts

 * Characteristic not implies fully invariant
 * Characteristic not implies fully invariant in finite abelian group
 * Characteristic not implies fully invariant in odd-order class two p-group

Opposite facts

 * Characteristic equals fully invariant in odd-order abelian group

For odd primes
Suppose $$p$$ is an odd prime. Let $$K$$ be the wreath product of groups of order p and $$G$$ be $$K/[[[K,K],K],K]$$. $$G$$ is thus a semidirect product of an elementary abelian group of order $$p^3$$ and a cyclic group of order $$p$$. Note that for $$p > 3$$, $$G$$ has exponent $$p$$ but for $$p = 3$$, $$G$$ has exponent $$9$$.

Consider $$H = C_G([G,G])$$, the centralizer of commutator subgroup. $$H$$ is a characteristic subgroup. Consider now an element $$x$$ of order $$p$$ outside $$H$$ (such an element exists because of the way the group is defined as a semidirect product) and a subgroup $$L$$ of $$G$$ of index $$p$$ other than $$H$$, that does not contain $$x$$. Consider the retraction from $$G$$ to $$\langle x \rangle$$ with kernel $$L$$. This retraction doesn't preserve $$H$$, so $$H$$ is not fully invariant.

For the prime two
Consider the dihedral group:

$$G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle$$.

Let $$H = \langle a \rangle$$. Then, $$H = C_G([G,G])$$, so $$H$$ is characteristic in $$G$$. Consider now the subgroup $$L = \langle a^2, x \rangle$$ and the element $$ax$$. Consider the retraction to the subgroup $$\langle ax \rangle$$ with kernel $$L$$. $$H$$ is not preserved under this retraction (for instance, $$a$$ gets mapped to $$x$$), so it is not fully invariant.