  We present a construction that yields infinite families of non-isomorphic
semidirect products $N \rtimes F_m$ sharing a specified profinite completion.
Within each family, $m \ge 2$ is constant and $N$ is a fixed group. For $m=2$
we can take $N$ to be free of rank $\ge 10$, free abelian of rank $\ge 12$, or
a surface group of genus $\ge 5$.
