If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?

Question

I recently answered the following question:

If a finitely generated semi-direct product $\mathbb{Z}$ acting on a non finitely generated group, can there be fixed points?

I have a related question:

Is there an example of a group $G=K\rtimes \mathbb{Z}$, such that

• $G$ is finitely generated but $K$ is not finitely generated
• the fixed points of $\phi(1)$, which is the automorphism on $K$ corresponding to $1_\mathbb{Z}$, are not a finitely generated group?

I suspect there is an example, but I don't have enough experience with these sorts of groups to come up with one right away.

EDIT: I've also asked this on Math Overflow, for which this is the link.

FURTHER EDIT: The question on Math Overflow has been answered positively.