Large orbits, projective Fraisse limits, and the pseudo-arc

We show that the conjugacy action of the automorphism group ${\rm Aut}(\mathbb{P})$, of the projective Fra\"{i}ss\'{e} limit $\mathbb{P}$, whose natural quotient is the pseudo-arc, on the set of involutions of $\mathbb{P}$, has a comeager orbit.