Lack of profinite rigidity among extensions with free quotient

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 $\ge 5$.