The setup is that $F$ is a free finitely generated group, $H, H'$ are subgroups of index $2$, and $\tau:H\to H'$ is an isomorphism.

Denote by $B_r$ the ball around $1$ of radius $r$ in $F$, in the word metric.

*Question:*

Is it possible to find a sequence $G_1,\dots, G_n$ of normal subgroups of $F$ of the same index, with $n$ depending on $r$, such that

a) $G_i\cap B_r=\{1\}$, for $i=1,\dots,n$

b) $G_i\not\subset H$ and $G_i\not\subset H'$ for $i=1,\dots,n$

c) $\tau(G_i\cap H)=G_{i+1}\cap H'$ for $i=1,\dots,n-1$

d) when $r\to\infty$ then $n\to\infty.$

I am interested in general in the dynamics of such partial automorphisms $\tau$ and the situation above is the one I would most like to understand.