Group of prime power order may have multiple characteristic subgroups of prime order

Statement
Let $$p$$ be a prime number. Then, there exists a group of prime power order $$P$$ that has more than one characteristic subgroup of order $$p$$.

Related facts

 * Maximal among abelian characteristic subgroups may be multiple and isomorphic
 * Characteristic maximal not implies isomorph-free in group of prime power order

Case $$p = 2$$
For $$p = 2$$, we can take:

$$P := \langle a,b,c \mid a^2 = b^4 = e, b^2 = c^2, ab = ba, ac = ca, cbc^{-1} = ab \rangle$$.

Then, $$P$$ has three characteristic subgroups of order two:


 * 1) The subgroup $$\langle a \rangle$$ is the commutator subgroup.
 * 2) The subgroup $$\langle b^2 \rangle$$ is the unique subgroup generated by the unique element that is a square but not a commutator.
 * 3) The subgroup $$\langle b^2a \rangle$$ is the unique subgrou pgenerated by the element that is a producto f squraes but not a square itself.