Omega subgroups not are variety-containing

Statement
We can have a group of prime power order (i.e., a finite $$p$$-group) $$P$$ such that the first omega subgroup $$\Omega_1(P)$$, defined as:

$$\Omega_1(P) := \langle x \mid x^p = e \rangle$$,

is not a variety-containing subgroup (i.e., fact about::variety-containing subgroup of group of prime power order) of $$P$$: there exists a subgroup $$H$$ of $$P$$ isomorphic to a subgroup of $$\Omega_1(P)$$ but that is not contained in $$\Omega_1(P)$$.

In particular, because of the equivalence of definitions of variety-containing subgroup of finite group, $$\Omega_1(P)$$ need not be a fact about::subisomorph-containing subgroup and it need not be a fact about::variety-containing subgroup.

Related facts

 * Omega subgroups are homomorph-containing

Proof
Suppose $$A$$ is a wreath product of groups of order p, i.e., $$A$$ is a group of order $$p^{p+1}$$ obtained as the semidirect product of an elementary abelian group of order $$p^p$$ by a cyclic group of order $$p$$ acting as automorphisms. $$A$$ is isomorphic to the $$p$$-Sylow subgroup of the symmetric group of degree $$p^2$$. In particular, $$A$$ has a cyclic subgroup of order $$p^2$$.

Suppose $$B$$ is a cyclic group of order $$p^2$$.

Define:

$$P := A \times B$$.

Then, $$\Omega_1(P) = A \times C$$ where $$C$$ is the subgroup of order $$p$$ in $$B$$. Consider the subgroup $$H = \{ e \} \times B$$. $$H$$ is isomorphic to a subgroup of order $$p^2$$ in $$A \times \{ e \}$$, which in turn is contained in $$\Omega_1(P)$$, but $$H$$ itself is not contained in $$\Omega_1(P)$$.