Descendance does not satisfy image condition

Statement
It is possible to have groups $$G$$ and $$K$$, a descendant subgroup $$H$$ of $$G$$ and a surjective homomorphism $$\varphi:G \to K$$ such that $$\varphi(H)$$ is not a descendant subgroup of $$K$$.

Example of the infinite dihedral group
We take:


 * $$G$$ to be the infinite dihedral group. We can take the presentation $$G = \langle a,x \mid xax = a^{-1}, x^2 = e \rangle$$.
 * $$H$$ to be a subgroup of order two complementary to the cyclic maximal subgroup, i.e., $$H$$ is a subgroup generated by a reflection. In the above presentation, we can take $$H = \langle x \rangle$$.
 * $$K$$ to be the quotient of $$G$$ by the subgroup generated by multiples of 3 in the cyclic maximal subgroup, with the natural quotient map. In symbols, $$K = G/\langle a^3 \rangle$$. $$K$$ is isomorphic to symmetric group:S3, and the image of $$H$$ under the quotient map corresponds to S2 in S3.

We can check that:


 * $$H$$ is descendant in $$G$$: It is the intersection of the following descending chain of subgroups, each of index two in its predecessor, hence each normal in its predecessor:

$$G = \langle a,x \rangle \ge \langle a^2,x \rangle \ge \langle a^4,x \rangle \ge \dots$$


 * The image of $$H$$ in $$K$$ is not descendant in $$K$$: The image looks like S2 in S3, which is a subgroup of a finite group, and is in fact a contranormal subgroup, so it cannot be descendant.