The covering radius of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(q)$

The covering radius of a subset $C$ of the symmetric group ${\mathrm{S}}_n$ is the maximal Hamming distance of an element of ${\mathrm{S}}_n$ from $C$. This note determines the covering radii of the finite projective general linear groups. It turns out that the covering radius of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(q)$ is $q-2$ if $q$ is even, and is $q-3$ if $q$ is odd.