TITLE: Howson property of automorphism group of $F_2$ and of $F_3$
QUESTION [7 upvotes]: Is the intersection of any two finitely generated subgroups of $\operatorname{Aut}(F_2)$ (resp. $\operatorname{Aut}(F_3)$) again finitely generated? That is, does $\operatorname{Aut}(F_2)$ (resp.  $\operatorname{Aut}(F_3)$) have the Howson property? Here $F_n$ is the free group of rank $n$.
For $n \geq 4$, it follows by @YCor’s comment that $\operatorname{Aut}(F_n)$ does not have the Howson property.
This question is maybe very simple (apologies if so!), but I have not been able to track down any references for this. I suppose my question extends quite naturally to the same question for the outer automorphism group $\operatorname{Out}(F_3)$ given the embedding $\operatorname{Aut}(F_n) \hookrightarrow \operatorname{Out}(F_{n+1})$. Note that $\operatorname{Out}(F_2) \cong \operatorname{GL}_2(\mathbb{Z})$, which is virtually free and hence has the Howson property.

REPLY [7 votes]: According to wikipedia (https://en.wikipedia.org/wiki/Howson_property), any group of the form $F_2 \rtimes \mathbb{Z}$ fails to have the Howson property. Assuming we believe wikipedia, then since $Aut(F_2)$ contains subgroups of this form, it also fails to have the Howson property. (For example, take the subgroup generated by $Inn(F_2)\cong F_2$ together with any $\alpha\in Aut(F_2)$ whose image in $Out(F_2)\cong GL_2(\mathbb{Z})$ has infinite order.)