Quotients by countable subgroups are hyperfinite

We show that for any Polish group $G$ and any countable normal subgroup $\mathrm{\Gamma }◃G$, the coset equivalence relation $G\mathrm{/}\mathrm{\Gamma }$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.