Omega subgroups not are prehomomorph-contained

Statement
It is possible to have a group of prime power order $$P$$ (i.e., a finite $$p$$-group $$P$$ for some prime number $$p$$), a subgroup $$K$$ of $$P$$, and a surjective homomorphism $$\varphi:K \to \Omega_1(P)$$, such that $$\Omega_1(P)$$ is not contained in $$K$$.

Analogous examples can be constructed for $$\Omega_k(P)$$ for any $$k \ge 1$$.

Example for $$k = 1$$
Let $$Q$$ be the quaternion group of order eight and $$C$$ be the cyclic group of order two. Define:

$$P := Q \times C$$

Let $$K = Q \times \{ e \}$$.

Then, $$K/Z(K) \cong Q/Z(Q)$$ which is a Klein four-group. Also, $$\Omega_1(P) = \Omega_1(Q) \times \Omega_1(C) = Z(Q) \times C$$ which is again a Klein four-group. Hence, $$K/Z(K) \cong \Omega_1(P)$$, so there is a surjective homomorphism \varphi:K \to \Omega_1(P). However, $$\Omega_1(P)$$ is not contained in $$K$$.