 Descent of an irreducible SL2 representation to PGL2

The irreducible rational representations of $S{L}_2(\mathbb{C})$ are ${\mathrm{Sym}}^n({\mathbb{C}}^2)$. Such a representation descends through $S{L}_2\to PG{L}_2$ exactly when the central element $-I$ acts trivially, which holds exactly when $n$ is even.