Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs

The Star graph $S_n$ is the Cayley graph of the symmetric group $\mathrm{Sym}_n$ with the generating set $\{(1\ i): 2\leq i\leq n \}$. Arumugam and Kala proved that $\{\pi\in \mathrm{Sym}_n: \pi(1)=1\}$ is a perfect code in $S_n$ for any $n$, $n\geq 3$. In this note we show that for any $n$, $n\geq 6$ the Star graph $S_n$ contains a perfect code which is the union of cosets of the embedding of $\mathrm{PGL}(2,5)$ into $\mathrm{Sym}_6$.