Characteristicity does not satisfy lower central series condition

Statement
It is possible to have a group $$G$$ and a characteristic subgroup $$H$$ of $$G$$ such that there is a positive integer $$k$$ for which the lower central series member $$\gamma_k(H)$$ is not a characteristic subgroup of the lower central series member $$\gamma_k(G)$$.

In fact, we can construct, for each positive integer $$k > 1$$, an example that works for that $$k$$. In the special case that $$k = 2$$, we obtain an example of a group $$G$$ and a characteristic subgroup $$H$$ of $$G$$ such that the derived subgroup $$H'$$ is not a characteristic subgroup of the derived subgroup $$G'$$.

Smallest order example for a 2-group
Suppose $$G$$ is the faithful semidirect product of E8 and Z4. We can think of it as the semidirect product of elementary abelian group:E8 (a three-dimensional vector space over field:F2) and cyclic group:Z4, where the generator of the latter acts as the following matrix on the former:

$$\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\\end{pmatrix}$$


 * $$\gamma_2(G) = G'$$, the derived subgroup of $$G$$, is a Klein four-group inside the base of the semidirect product, generated by the first two basis vectors.
 * Let $$H = C_G(G')$$. $$H$$ contains the base of the semidirect product as well as the square of the generator of cyclic group:Z4. Explicitly, $$H$$ is the semidirect product of the base by the element acting as follows:

$$\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$$
 * $$H$$ is isomorphic to direct product of D8 and Z2. The derived subgroup $$\gamma_2(H) = H'$$ of $$H$$ is isomorphic to cyclic group:Z2 (generated by the second basis vector) and it lives inside $$G'$$ as Z2 in V4. Thus, $$H'$$ is not a characteristic subgroup of $$G'$$.

Smallest order example for a 3-group
Let $$G$$ be the group $$\mathbb{Z}_3 \wr \mathbb{Z}_3$$. Explicitly, $$G$$ is the semidirect product of elementary abelian group:E27, viewed as a three-dimensional vector space over field:F3, with the acting element of order three acting by the matrix:

$$\begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\\end{pmatrix}$$


 * $$\gamma_2(G) = G'$$ is the subgroup in the base comprising the elements whose coordinates add up to zero. It is an elementary abelian group of prime-square order.
 * $$H$$ is defined as the internal semidirect product of $$G'$$ and any element of three outside the base. There is a unique possibility for such a subgroup $$H$$, and it is isomorphic to unitriangular matrix group:UT(3,3).
 * $$\gamma_2(H) = H'$$ is a subgroup of order 3. It lives as a group of order 3 inside $$G'$$ which is elementary abelian of order 9, hence is not characteristic in $$G'$$.

Example for larger $$k$$
For a positive integer $$k$$, choose $$p$$ to be a prime number greater than $$k$$.


 * Define $$G$$ as the wreath product of groups of prime order $$\mathbb{Z}_p \wr \mathbb{Z}_p$$. This is a group of order $$p^{p+1}$$ with an elementary abelian normal subgroup of order $$p^p$$ and a complement of order $$p$$ acting by cyclic permutation of the coordinates.
 * Define $$H$$ as the internal semidirect product of $$G'$$ with an element of order $$p$$ outside the elementary abelian normal subgroup.
 * $$\gamma_k(H)$$ and $$\gamma_k(G)$$ are both elementary abelian subgroups of $$G$$, with $$\gamma_k(H)$$ having order $$p^{p-k}$$ and $$\gamma_k(G)$$ having order $$p^{p-k+1}$$. In particular, $$\gamma_k(H)$$ is a proper nontrivial subgroup of the elementary abelian $$p$$-group $$\gamma_k(G)$$, hence it is not characteristic inside $$\gamma_k(G)$$.