An involution on a pair of pants fixing one boundary component and permuting other two?

Question

Let $S$ be a pair of pants (i.e. sphere with 3 boundary components). Let $\gamma_1, \gamma_2, \gamma_3$ denote the 3 boundary components of $S$. Does there exist an orientation-preserving homeomorphism $\phi:S\rightarrow S$ such that $\phi^2=\mathrm{Id}$, $\phi(\gamma_1)=\gamma_2$ and $\phi|_{\gamma_3}=\mathrm{Id}$? .